0,2

Number of partitions of n into parts but there are two kinds of parts of size one.

Also number of graphical forest partitions of 2n+2.

a(n) = count 2 for each partition of n and 1 for each decrement. E.g. the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2+3+2+3+2=12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry (perry(AT)globalnet.co.uk), Feb 06 2004

Also the total number of all different integers in all partitions of n. E.g. a(4)=7 because the partition of n=4 comprises the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder (wieder.thomas(AT)t-online.de), Apr 10 2004

Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g. for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder (wieder.thomas(AT)t-online.de), Mar 08 2005

With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).

Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley (se16(AT)btinternet.com), Aug 01 2005

Equals left border of triangle A137633 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 31 2008

Equals row sums of triangle A027293 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]

Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 09 2008]

Equals A036469 convolved with A035363. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 2009]

C. C. Cadogan, On partly ordered partitions of a positive integer, Fib. Quart., 9 (1971), 329-336.

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

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.

A. M. Odlyzko, Asymptotic Enumeration Methods, p. 19

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

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).

T. D. Noe, Table of n, a(n) for n=0..1000

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

P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998)

D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, Ars Combinatoria, Vol. 65 (2002), 33-37.

H. Fripertinger, Partitions of an Integer

N. Hobson, Nick's Mathematical Puzzles, Partition identity (or a proof of Stanley's Theorem)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 113

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 126

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

F. Ruskey, Combinatorial Objects Server

N. J. A. Sloane, Transforms

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

Euler transform of 2 1 1 1 1 1 1...

log(a(n)) ~ -3.3959 + 2.44613*Sqrt(n). - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 11 2002

a(n)=1/n*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 22 2002

G.f.: (1/(1-x))*Product(1/(1-x^m)), m=1..inf.

Sequence seems to have the same parity as A027349. Comment from James Sellers, Mar 08 2006: that is true.

a(n) =(1+O(n^(-1/6)))/(2*C*sqrt(3n))*exp(C*sqrt(n)) where C=Pi*sqrt(2/3) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 31 2004

a(n) =A000041(2n+1)-A110618(2n+1) =A000041(2n+2)-A110618(2n+2) - Henry Bottomley (se16(AT)btinternet.com), Aug 01 2005

Row sums of triangle A133735 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 22 2007

a:=n->(sum((numbpart(j)), j=0..n)):seq(a(n), n=0..44); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]

CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (from Robert G. Wilson v Jul 13 2004)

(* First do *) Needs["Combinatorica`"] (* then *) Table[ Count[ Flatten@ Partitions@ n, 1], {n, 45}] (from Robert G. Wilson v, (rgwv(AT)rgwv.com) Aug 06 2008)

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

Row sums of triangle A092905.

Cf. A014153, A024786, A026905, A058884, A093694, A133735, A137633.

A027293 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]

A010815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 09 2008]

A036469, A035363 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 2009]

Sequence in context: A079719 A036439 A035298 this_sequence A008609 A100823 A102346

Adjacent sequences: A000067 A000068 A000069 this_sequence A000071 A000072 A000073

nonn,easy,nice

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