0,3

Also number of nonnegative solutions to b+2c+3d+4e+...=n and the number of nonnegative solutions to 2c+3d+4e+...<=n. - Henry Bottomley (se16(AT)btinternet.com), Apr 17 2001

a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).

Also the number of rooted trees with n+1 nodes and height at most 2.

Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras gl(n). A006950, A015128 and this sequence together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003

a(n)=a(0)b(n)+a(1)b(n-2)+a(2)b(n-4)+... where b=A000009.

Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 16 2004

Number of graphs on n vertices that do not contain P3 as an induced subgraph. - Washington Bomfim (webonfim(AT)bol.com.br), May 10 2005

It is unknown if there are infinitely many partition numbers divisible by 3, although it is known that there are infinitely many divisible by 2. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 21 2005

Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 07 2005

a(n) = A114099(9*n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 15 2006

Comment from Maurice D. Craig (towenaar(AT)optusnet.com.au), Oct 30 2006: sequence agrees with expansion of Molien series for symmetric group S_n up to the term in x^n.

Also the number of nonnegative integer solutions to x_1+x_2+x_3+...+x_n=n such that n>=x_1>=x_2>=x_3>=...>=x_n>=0, because by letting y_k=x_k-x_(k+1)>=0 (where 0<k<n) we get y_1+2y_2+3y_3+...+(n-1)y_(n-1)+nx_n=n. - Werner Grundlingh (wgrundlingh(AT)gmail.com), Mar 14 2007

Let P(z):= Sum{j=0..inf} b_j z^j, b_0 != 0. Then 1/P(z) = Sum{j=0..inf} c_j z^j, where the c_j must be computed from the infinite triangular system b_0 c_0 = 1, b_0 c_1 + b_1 c_0 = 0 and so on (Cauchy products of the coefficients set to zero). The n-th partition number arises as the number of terms in the numerator of the expression for c_n: The coefficient c_n of the inverted power series is a fraction with b_0^(n+1) in the denominator and in its numerator having a(n) products of n coefficients b_i each. The partitions may be read off from the indices of the b_i. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007

A026820(a(n),n) = A134737(n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 07 2007

Equals row sums of triangle A137683 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 05 2008

This is also the number of parts equal to 1 in the outer shell of the partitions of n+1 (see A138151). - Omar E. Pol (info(AT)polprimos.com), Apr 17 2008

a(n)= the number of different ways to run up a staircase with n steps, taking steps of sizes 1,2,3,... and r (r<=n), where the order is not important and there is no restriction on the number or the size of each step taken. - Mohammad K. Azarian (azarian(AT)evansville.edu), May 21 2008

Equals the eigenvector of triangle A145006 and row sums of the eigentriangle of the partition numbers, A145007. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 28 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008: (Start)

Starting with offset 1 = INVERT transform of (1, 1, 0, 0, -1, 0, -1,...),

where A080995, the characteristic function of A001318 (1, 2, 5, 7, 12,...) is

signed (++ -- ++,...) as to 1's. This is equivalent to Lim__{n=1..inf}

A145006^n as a vector. The INVERT transform of (1, 1, 0, 0, -1,...) begins (1, 2,..)

then for each successive operation we take a dot product of (1, 1, 0, 0, -1,...) in reverse and the ongoing results of our series (1, 2, 3, 5, 7,...)

then add the result to the next term in (1, 1, 0, 0, -1,...). For example, a

(7) = 15 = (0, -1, 0, 0, 1, 1) dot (1, 2, 3, 5, 7, 11) = (0*1, (-1)*2, 0*3, 0*5, 1*7, 1*11)

= (-2 + 7 + 11) = 16, then add to (-1) = 15. (End)

Convolved with A147843 = A000203 prefaced with a zero: (0, 1, 3, 4, 7,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2009: (Start)

Equals an infinite convolution product_(1,1,1,...)*(1,0,1,0,1,...)*

(1,0,0,1,0,0,1,...)*(1,0,0,0,1,0,0,0,1,...)* ...; = a*b*c*...; where a =

(1/(1-x), b = (1/(1-x^2), c = (1/(1-x^3), ...etc. An array by rows: row 1 =

a, row 2 = a*b, row 3 = a*b*c,...; gives:

1, 1, 1, 1, 1, 1,. 1,. 1,. 1,..1,... = (a).................................

1, 1, 2, 2, 3, 3,. 4,..4,. 5,..5,... = (a*b)...............................

1, 1, 2, 3, 4, 5,. 7,..8,.10,.11,... = (a*b*c).............................

1, 1, 2, 3, 4, 5,. 6,..9,.11,.17,... = (a*b*c*d)...........................

1, 1, 2, 3, 5, 5,. 7,.10,.13,.18,... = (a*b*c*d*e).........................

1, 1, 2, 3, 5, 7,.11,.14,.20,.25,... = (a*b*c*d*e*f).......................

1, 1, 2, 3, 5, 7,.11,.15,.21,.27,... = (a*b*c*d*e*f*g).....................

1, 1, 2, 3, 5, 7,.11,.15,.22,.28,... = (a*b*c*d*e*f*g*h)...................

1, 1, 2, 3, 5, 7,.11,.15,.22,.29,... = (a*b*c*d*e*f*g*h*i).................

...with rows tending to A000041. Partition triangles A058398 = ascending

antidiagonals. Partition triangle A008284 reversal of A058398. (End)

a(n) is also the number of partitions of 2n into even parts. More generally, it appears that a(n) is also the number of partitions of k*n into parts divisible by k, for k>0 and n>0. [From Omar E. Pol (info(AT)polprimos.com), Nov 20 2009]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976

G. E. Andrews & K. Ericksson, Integer Partitions, Cambridge University Press 2004.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 307.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III.

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997

B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.

L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp. 101-164,Chelsea NY 1992.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 37, Eq. (22.13).

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.

G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-.

J. M. Kane, Distribution of orders of Abelian groups, Math. Mag., 49 (1976), 132-135.

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4.

S. Markovski and M. Mihova, An explicit formula for computing the partition numbers p(n), preprint, 2005

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.1, p. 491.

S. Ramanujan, Collected Papers, Chap. 25, Cambridge Univ. Press 1927 (Proceedings of the Camb.Phil.Soc., 19(1919)207-213).

S. Ramanujan, Collected Papers, Chap. 28, Cambridge Univ. Press 1927 (Proceedings of the London Math.Soc., 2, 18(1920)).

S. Ramanujan, Collected Papers, Chap. 30, Cambridge Univ. Press 1927 (Mathematische Zeitschrift, 9(1921)147-163).

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 122.

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.

J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. E. Tapscott and D. Marcovich, "Enumeration of Permutational Isomers: The Porphyrins", Journal of Chemical Education, 55 (1978), 446-447. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Sep 21 2008]

Robert M. Young, "Excursions in Calculus", Mathematical Association of America, p. 367. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008]

Robert M. Ziff, "On Cardy's formula for the critical crossing probability in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).

David W. Wilson, Table of n, a(n) for n = 0..10000

Joerg Arndt, Fxtbook

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

S. Ahlgren and K. Ono, Addition and Counting: The Arithmetic of Partitions

S. Ahlgren & K. Ono, Congruence properties for the partition function

S. Ahlgren & K. Ono, Congruence properties for the partition function

G. Almkvist, Asymptotic Formulas and Generalized Dedekind Sums

G. Almkvist and H. S. Wilf, On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n)

Amazing Mathematical Object Factory, Information on Partitions [Broken link corrected by Steve Vonn (5463math(AT)gmail.com), Jan 03 2009]

G. E. Andrews, Three Aspects of Partitions

G. E. Andrews, On a Partition Function of Richard Stanley.

G. E. Andrews & K. Ono, Ramanujan's congruences and Dyson's crank

G. E. Andrews & R. Roy, Ramanujan's Method in q-series Congruences

Anonymous, Bibliography on Partitions

A. O. L. Atkins & F. G. Garvan, Relations between the ranks and cranks of partitions

A. Berkovich & F. G. Garvan, On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5

A. Berkovich & F. G. Garvan, On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations

B. C. Berndt, Ramanujan's congruences for the partition function modulo 5,7 and 11

B. C. Berndt and K. Ono, Ramanujan's Unpublished Manuscript On The Partition And Tau Functions With Proofs And Commentary

B. C. Berndt and K. Ono, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

H. Bottomley, Illustration of initial terms

H. Bottomley, Illustration of initial terms of A000009, A000041 and A047967

H. Bottomley, Partition and composition calculator

K. S. Brown, Additive Partitions of Numbers

K. S. Brown's Mathpages, Computing the Partitions of n

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions.

J. Davis & E. Perez, Computations Of The Partition Function, p(n)

N. J. Fine, Some New Results On Partitions

B. Forslund, Partitioning Integers

H. Fripertinger, Partitions of an Integer

GEO magazine, Zahlenspalterei

A. Hassen and T. J. Olsen, Playing With Partitions On The Computer

A. D. Healy, Partition Identities

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 61

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 74

E. Klarreich, Pieces of Numbers: A proof brings closure to a dramatic tale of partitions and primes, Science News, Week of Jun 18, 2005; Vol. 167, No. 25, p. 392.

J. Laurendi, Partitions of Integers

T. Lockette, Explore Magazine, "Path To Partitions"

Dr. Math, Partitioning the Integers

Dr. Math, Partitioning an Integer

M. MacMahon, Collected Papers of Ramanujan, Table for p(n);n=1 through 200

G. P. Michon, Table of partition function p(n) (n=0 through 4096)

G. P. Michon, Partition function

G. A. Miller, Number Of The Abelian Groups Of A Given Order

Hisanori Mishima, Factorization of Partition Numbers

D. J. Newman, A Simplified Proof Of The Partition Formula

K. Ono, Arithmetic of The Partition Function

K. Ono, Parity Of The Partition Function

K. Ono, Distribution of the partition function modulo m

T. J. Osler, Playing with Partitions on the Computer

I. Peterson, The Power Of Partitions

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

M. Planat, Quantum 1/f Noise in Equilibrium: from Planck to Ramanujan

S. Plouffe, Partitions [Contains first 10000000 terms]

S. Plouffe, Partition numbers through n = 300000

S. Plouffe, Partitions numbers from 300000 to 450000

S. Plouffe, Partitions numbers from 450000 to 500000

O. E. Pol, How to build a shell model of partitions [From Omar E. Pol (info(AT)polprimos.com), Sep 06 2008]

O. E. Pol, A shell model of partitions (2D and 3D) [From Omar E. Pol (info(AT)polprimos.com), Sep 06 2008]

O. E. Pol, Illustration of initial terms (2D view) [From Omar E. Pol (info(AT)polprimos.com), Sep 06 2008]

O. E. Pol, Illustration of initial terms (3D view) [From Omar E. Pol (info(AT)polprimos.com), Sep 06 2008]

M. Presern, Some Results On Partitions

W. A. Pribitkin, The Ramanujan Journal 4(4) 2000, Revisiting Rademacher's Formula for the Partition Function p(n)

PYTHAGORAS, Ramanujan and The Partition Function(Text in Dutch)

S. Ramanujan, Some Properties Of p(n), The Number Of Partitions Of n

S. Ramanujan, Congruence Properties Of Partitions

S. Ramanujan, Congruence Properties Of Partitions

S. Ramanujan & G. H. Hardy, Une formule asymptotique pour le nombre de partitions de n

J. D. Rosenhouse, Partitions of Integers

J. D. Rosenhouse, Solutions to Problems

F. Ruskey, Generate Numerical Partitions

F. Ruskey, The first 284547 partition numbers (52MB compressed file)

M. Savic, The Partition Function and Ramanujan's 5k+4 Congruence

T. Sillke, Number of integer partitions

R. P. Stanley, A combinatorial miscellany

R. L. Weaver, The Ramanujan Journal 5(1) 2001, New Congruences for the Partition Function

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics(3)

West Sussex Grid for Learning, Multicultural Mathematics, Ramanujan's Partition of Numbers

Thomas Wieder, Comment on A000041

Wikipedia, Integer Partition

H. S. Wilf, Lectures on Integer Partitions

Wolfram Research, Generating functions of p(n)

D. J. Wright, Partitions

Index entries for "core" sequences

Index entries for related partition-counting sequences

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Index entries for sequences related to rooted trees

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 41

G.f.: Product_{k>0} 1/(1-x^k) = Sum_{k>= 0} x^k Product_{i = 1..k} 1/(1-x^i) = 1+Sum_{k>0} x^(k^2)/(Product_{i = 1..k} (1-x^i))^2.

a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + ... = 0, where the sum is over n-k and k is a generalized pentagonal number (A001318) <= n and the sign of the k-th term is (-1)^([(k+1)/2]). See A001318 for a good way to remember this!

a(n) = (1/n) * Sum_{k=0, 1, ..., n-1} sigma(n-k)*a(k), where sigma(k) is the sum of divisors of k (A000203).

a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy and Ramanujan).

a(n) < exp( (2/3)^(1/2) pi sqrt(n) ) (Ayoub, p. 197).

G.f.: Product (1+x^m)^A001511(m); m=1..inf. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 26 2004

a(n)=sum(i=0, n-1, P(i, n-i)), where P(x, y) is the number of partitions of x into at most y parts and P(0, y)=1. - Jon Perry (perry(AT)globalnet.co.uk), Jun 16 2003

G.f. : product(i=1, oo, product(j=0, oo, (1+x^((2i-1)*2^j))^(j+1))) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004

G.f. e^{Sum_{k>0} (x^k/(1-x^k)/k)}. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 08 2006

Euler transform of all 1's sequence (A000012). Weighout transform of A001511. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 15 2006

a(n) = A027187(n)+A027193(n) = A000701(n)+A046682(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006

Row sums of triangles A133734 and A133736. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 22 2007

1/(1-x) + sum{ k=1,oo, x^k(k+1)/ prod(i=1,k, (1-x^i)^2)*(1-x^k+1) } (the pronic equivalent of the Durfee Square GF) [From Jon Perry (johnandruth(AT)jrperry.orangehome.co.uk), Aug 02 2008]

Convolved with A152537 gives A000079, powers of 2. [From Gary W. Adamson (qntmpkt(AT)haoo.com), Dec 06 2008]

with(combinat); A000041 := numbpart; [ seq(numbpart(i), i=0..50) ]; [Warning: Maple 10 and 11 give incorrect answers in some cases, for example combinat[numbpart](11269); is wrong.]

spec := [ B, {B=Set(Set(Z, card>=1))}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=0..50)];

with(combstruct):ZL0:=[S, {S=Set(Cycle(Z, card>0))}, unlabeled]:seq(count(ZL0, size=n), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2007

G:={P=Set(Set(Atom, card>0))}:combstruct[gfsolve](G, labeled, x); seq(combstruct[count]([P, G, unlabeled], size=i), i=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2007

Table[ PartitionsP[n], {n, 0, 45}]

(MAGMA) a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];

(PARI) a(n)=if(n<0, 0, polcoeff(1/eta(x+x*O(x^n)), n))

(PARI) The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): - Ralf Stephan (ralf(AT)ark.in-berlin.de), Nov 30 2002

Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))

L(n, q) = if(q==1, 1, sum(h=1, q-1, if(gcd(h, q)>1, 0, cos((g(h, q)-2*h*n)*Pi/q))))

g(h, q) = if(q<3, 0, sum(k=1, q-1, k*(frac(h*k/q)-1/2)))

part(n) = round(sum(q=1, max(5, 0.24*sqrt(n)+2), L(n, q)*Psi(n, q)))

(PARI) a(n)=numbpart(n)

(PARI) a(n)=if(n<0, 0, polcoeff(sum(k=1, sqrtint(n), x^k^2/prod(i=1, k, 1-x^i, 1+x*O(x^n))^2, 1), n))

(PARI) f(n)= {local(v, i, k, s, t); v=vector(n, k, 0); v[n]=2; t=0; while(v[1]<n, i=2; while(v[i]==0, i++); v[i]--; s=sum(k=i, n, k*v[k]); while(i>1, i--; s+=i*(v[i]=(n-s)\i)); t++); t } (Thomas Baruchel (baruchel(AT)users(AT)sourceforge.net), Nov 07 2005)

(Mupad) combinat::partitions::count(i) $i=0..54 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 16 2007

(Other) sage: [number_of_partitions(n)for n in xrange(0, 46)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 24 2009]

Cf. A000009, A008284, A008284, A000203, A001318.

For successive differences see A002865, A053445, A072380, A081094, A081095.

Antidiagonal sums of triangle A092905.

Cf. A132311.

Cf. A138151.

Cf. A135010, A138121. [From Omar E. Pol (info(AT)polprimos.com), Sep 06 2008]

A145006, A145007 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 28 2008]

A080995 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008]

A147843 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]

Cf. A152537, A000079 [From Gary W. Adamson (qntmpkt(AT)haoo.com), Dec 06 2008]

Sequence in context: A008641 A046054 A092885 this_sequence A084251 A024794 A091955

a(n) = A035363(2n). [From Omar E. Pol (info(AT)polprimos.com), Nov 20 2009]

Adjacent sequences: A000038 A000039 A000040 this_sequence A000042 A000043 A000044

core,easy,nonn,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Links contributed by Patrick De Geest (pdg(AT)worldofnumbers.com), Oct 15 1999. Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001 and from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001.

Further links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com), Spring 2003