Search: id:A000219 Results 1-1 of 1 results found. %I A000219 M2566 N1016 %S A000219 1,1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297, %T A000219 18334,29601,47330,75278,118794,186475,290783,451194,696033,1068745, %U A000219 1632658,2483234,3759612,5668963,8512309,12733429,18974973,28175955 %N A000219 Number of planar partitions of n. %C A000219 Two-dimensional partitions of n in which no row or column is longer than the one before it (compare A001970). E.g. a(4) = 13: %C A000219 4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2 %C A000219 .....1....2.....1...1......1...11.1..1........ 11 %C A000219 ....................1.............1..1 %C A000219 .....................................1 %C A000219 Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner - Wouter Meeussen (wouter.meeussen(AT)pandora.be). %C A000219 Also number of partitions of n objects of 2 colors, each part containing at least one black object. - (Christian G. Bower (bowerc(AT)usa.net), Jan 08 2004) %C A000219 Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., k types of part k. e.g. n=3 gives 111, 12, 12', 3, 3', 3''. - Jon Perry (perry(AT)globalnet.co.uk), May 27 2004 %C A000219 Can also be regarded as the number of Jordan canonical forms for an nxn matrix. (i.e. a 5x5 matrix has 24 distinct Jordan canonical forms, dependent on the algebraic and geometric multiplicity of each eigenvalue.) [From Aaron Gable (agable(AT)hmc.edu), May 26 2009] %C A000219 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009: (Start) %C A000219 1/n * convolution product of n terms * A001157 (sum of squares of divisors of n): (1, 5, 10, 21, 26, 50, 50, 85,...) = a(n). As shown by [Bressoud, p.12]: 1/6 * [1*24 + 5*13 + 10*6 + 21*3 + 26*1 + 50*1] = 288/6 = 48. %C A000219 Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13,...) = A026007: (1, 1, 2, 5, 8, 16, 28, 49, 83,...). (End) %C A000219 Starting with offset 1 = row sums of triangle A162453 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2009] %C A000219 Also number of functions from an n-set to itself, up to permutation of the set (compare to partition function, which is number of conjugacy classes in S_n) [From Harry Altman (haltman(AT)umich.edu), Nov 21 2009] %D A000219 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359. %D A000219 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241. %D A000219 A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. %D A000219 Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 1972. %D A000219 Bender, E. A. and Knuth, D. E. ``Enumeration of Plane Partitions.'' J. Combin. Theory A. 13, 40-54, 1972. %D A000219 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10. %D A000219 D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646. %D A000219 L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6). %D A000219 D. E. Knuth, A Note on Solid Partitions. Math. Comp. 24, 955-961, 1970. %D A000219 P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373. %D A000219 P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332. %D A000219 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000219 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000219 E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505. %H A000219 T. D. Noe, Table of n, a(n) for n = 0..400 %H A000219 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359. %H A000219 G. E. Andrews, P. Paule, MacMahon's partition analysis XII: Plane Partitions, J. Lond. Math. Soc., 76 (2007), 647-666. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 19 2009] %H A000219 Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 18 %H A000219 H. Bottomley, Illustration of initial terms %H A000219 P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580 %H A000219 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 141 %H A000219 P. A. MacMahon, Combinatory analysis. %H A000219 Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ... %H A000219 L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions... %H A000219 N. J. A. Sloane, Transforms %H A000219 J. Stienstra, Mahler measure, Eisenstein series and dimers %H A000219 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000219 Index entries for "core" sequences %F A000219 G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912. %F A000219 Euler transform of sequence [1, 2, 3, ...]. %F A000219 a(n) ~ (c_2 / n^(25/26)) * exp( c_1 * n^(2/3) ), where c_1 = 2.00945... and c_2 = 0.40099... - Wright, 1931. %F A000219 a(n)=1/n*Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n)=A001157(n)=sum of squares of divisors of n. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 20 2002 %F A000219 G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 10 2003 %e A000219 A planar partition of 13: %e A000219 4 3 1 1 %e A000219 2 1 %e A000219 1 %e A000219 a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+15*sigma_2(1)*sigma_2(2)^2+\ 30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24*sigma_2(5)) = 24. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 10 2003 %p A000219 series(mul((1-x^k)^(-k),k=1..64),x,63); %t A000219 Rest@CoefficientList[ Series[ Product[ (1-x^k)^-k, {k, 1, 64} ], {x, 0, 64} ], x ] %o A000219 (PARI) a(n)=if(n<0,0,polcoeff(exp(sum(k=1,n,x^k/(1-x^k)^2/k,x*O(x^n))), n)) /* Michael Somos Jan 29 2005 */ %o A000219 (PARI) a(n)=if(n<0,0,polcoeff(prod(k=1,n,(1-x^k+x*O(x^n))^-k),n)) /* Michael Somos Jan 29 2005 */ %o A000219 Contribution from R. J. Mathar (mathar(AT)st.rw.leidenuniv.nl), Oct 18 2009: (Start) %o A000219 (Python) def divisors(n): %o A000219 ...a = {1,n} %o A000219 ...for x in range(2,n//2+1): %o A000219 ......if n % x == 0: %o A000219 .........a |= {x} %o A000219 ...return a %o A000219 def sigma(n,k): %o A000219 ...a=0 %o A000219 ...for d in divisors(n): %o A000219 ......a += d**k %o A000219 ...return a %o A000219 def A000219(n): %o A000219 ...if n <=1: %o A000219 ......return 1 %o A000219 ...else: %o A000219 ......a=0 %o A000219 ......for k in range (1,n+1): %o A000219 .........a += A000219(n-k)*sigma(k,2) %o A000219 ......return a//n %o A000219 print([A000219(n) for n in range(0,20)]) (End) %Y A000219 Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300. %Y A000219 Cf. A023871-A023878. %Y A000219 Row sums of A089353 and A091438. %Y A000219 Cf. A026007, A001157. %Y A000219 A162453 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2009] %Y A000219 Sequence in context: A018081 A001452 A005405 this_sequence A027999 A005196 A032287 %Y A000219 Adjacent sequences: A000216 A000217 A000218 this_sequence A000220 A000221 A000222 %K A000219 nonn,nice,easy,core,new %O A000219 0,3 %A A000219 N. J. A. Sloane (njas(AT)research.att.com). %E A000219 Corrected Jul 29 2006 Search completed in 0.002 seconds