A000293 -id:A000293 - cp's OEIS frontend

A007326 Difference between A000294 and the number of solid partitions of n (A000293).

0, 0, 0, 0, 0, 0, 1, 3, 8, 19, 40, 83, 176, 365, 775, 1643, 3483, 7299, 15170, 31010, 62563, 124221, 243296, 469856, 896491, 1690475, 3155551, 5834871, 10701036, 19479021, 35227889, 63335778, 113286272, 201687929, 357585904, 631574315, 1111614614, 1950096758, 3410420973, 5946337698, 10337420278, 17918573379, 30968896662, 53366449357, 91689380979, 157058043025, 268210414468, 456613323892

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Understanding this sequence is a famous unsolved problem in the theory of partitions.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..72
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.
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. [Annotated scanned copy]

a(n) = A000294(n) - A000293(n).

Cf. A007327, A007328, A007329, A007330, A008780, A042984.

Entry revised by Sean A. Irvine and N. J. A. Sloane, Dec 18 2017

A037452 Let F(x) = 1 + 1x + 4x^2 + 10*x^3 + ..., the g.f. for A000293 (solid partitions), and write F(x) = 1/Product_{n>=1} (1 - x^n)^a(n).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 20, 26, 34, 46, 68, 97, 120, 112, 23, -186, -496, -735, -531, 779, 3894, 9323, 16472, 23056, 23850, 10116, -31613, -120720, -283202, -548924, -932162, -1380125, -1655072, -1144651, 1385629, 7943203, 21083967, 42787785, 71816191, 98995196, 100392874, 29623771, -187433150

Offset: 0

N. J. A. Sloane, May 02 2003

sign

Suresh Govindarajan, Table of n, a(n) for n = 0..72
Suresh Govindarajan, Exact Enumeration of Solid Partitions, IIT Madras.
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.

Cf. A000293, A002835, A080207, A193718.

More terms from Vladeta Jovovic, Jul 30 2003

A002836 Let F(x) = 1 + x + 4x^2 + 10x^3 + ... = g.f. for A000293 (solid partitions) and expand (1-x)(1-x^2)(1-x^3)...*F(x) in powers of x.

Original entry on oeis.org

1, 0, 2, 5, 12, 24, 56, 113, 248, 503, 1043, 2080, 4169, 8145, 15897, 30545, 58402, 110461, 207802, 387561, 718875, 1324038, 2425473, 4416193, 7999516, 14411507, 25837198, 46092306, 81851250, 144691532, 254682865, 446399687, 779302305

Offset: 0

N. J. A. Sloane

nonn

Convolved with A000041 = A000293, solid partitions; and left border of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

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).

Suresh Govindarajan, Table of n, a(n) for n = 0..72
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
Physics enthusiasts at IIT Madras, The Solid Partitions Project

Cf. A000293, A005980, A000293, A000041, A161564.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

More terms from Srivatsan Balakrishnan and Suresh Govindarajan, Jan 03 2013

A161564 Convolution triangle for A000293 (solid partitions): A000041 * A002836.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 5, 2, 0, 3, 12, 5, 4, 0, 5, 24, 12, 10, 6, 0, 7, 56, 24, 24, 15, 10, 0, 11, 113, 56, 48, 36, 25, 14, 0, 15, 248, 113, 112, 72, 60, 35, 22, 0, 22, 503, 248, 226, 168, 120, 84, 55, 30, 0, 30

Offset: 0

Gary W. Adamson, Jun 13 2009

Row sums = A000293, solid partitions: (1, 1, 4, 10, 26, 59, 140, 307, 684,...)

			First few rows of the triangle =
1;
0, 1;
2, 0, 2;
5, 2, 0, 3;
12, 5, 4, 0, 5;
24, 12, 10, 6, 0, 7;
56, 24, 24, 15, 10, 0, 11;
113, 56, 48, 36, 25, 14, 0, 15;
248, 113, 112, 72, 60, 35, 22, 0, 22;
503, 248, 226, 168, 120, 84, 55, 30, 0, 30;
...
Example: row 4 = (12, 5, 4, 0, 5), sum = 26 = A000293(4).

Cf. A000293, A000041, A002836

Triangle read by rows, A000041 convolved with A002836: (1, 0, 2, 5, 12, 24, 56,...). Antidiagonals of a convolution array, A002836 * A000041: 1, . 1, . 2, . 3, . 5, . 7, . 11,... 0, . 0, . 0, . 0, . 0, . 0, .. 0,... 2, . 2, . 4, . 6, .10, .14, ..22,... 5, . 5, . 10, .15, .25, .35, ..55,... 12 .12, . 24, .36, .60, .84, .132,... ...

A193718 Let F(x) = 1 + 1x + 4x^2 + 10x^3 + ..., the g.f. for A000293 (solid partitions), and write F(x) = 1/Product_{n>=1} E(n)^a(n) where E(n) = Product_{k>=n} (1 - x^(nk)).

Original entry on oeis.org

1, 2, 5, 7, 14, 12, 25, 24, 40, 51, 96, 93, 111, -5, -206, -530, -736, -591, 778, 3819, 9292, 16373, 23055, 23706, 10101, -31727, -120766, -283232, -548925, -932041, -1380126, -1654576, -1144753, 1386362, 7943163, 21084398, 42787784, 71815410, 98995079, 100388956, 29623770, -187442482, -648478235, -1449118398, -2615085854, -3963369427, -4855203952, -3819950381, 1908741941, 16724652946

Offset: 0

Joerg Arndt, Aug 03 2011

sign

Cf. A037452 (expansion 1/Product_{n>=1} (1 - x^n)^a(n)), A000293 (solid partitions).

A277613 Logarithmic derivative of the g.f. of the solid partitions A000293.

Original entry on oeis.org

1, 7, 19, 47, 76, 145, 183, 319, 433, 762, 1068, 1625, 1457, 511, -2696, -7617, -12494, -8999, 14802, 78682, 195984, 363458, 530289, 574297, 252976, -820475, -3259007, -7929105, -15918795, -27966750, -42783874, -52969921, -37772397, 47098898, 278012363, 759015293, 1583148046, 2729030066, 3860814119, 4015793914, 1214574612, -7871995868, -27884564061, -63760120938, -117678872282, -182313402679, -228194585696, -183355932567, 93528356566, 836233409412, 2360489258476, 4956621402741, 8577450776595, 12176709992155, 12572248705543, 2874527812671, -29026344726969, -100513507605919, -229939345736773, -423043591887710, -643162163240861, -757839109104688, -458886747576888, 831588355306815, 4020413344163097, 10249469548463477, 20417504944664974, 33937902760293134, 46224437161712292, 44445354551818961, 1635692222011481, -129140996172417587

Offset: 1

Paul D. Hanna, Nov 20 2016

sign

Based on the solid partitions calculated by Suresh Govindarajan and listed in A000293.

Finding a formula for this sequence is an unsolved problem; at first it was thought to be A278403, where: Sum_{n>=1} A278403(n)*x^n/n = log( Product_{n>=1} 1/(1 - x^n)^(n*(n+1)/2) ).

			L.g.f.: L(x) = x + 7*x^2/2 + 19*x^3/3 + 47*x^4/4 + 76*x^5/5 + 145*x^6/6 + 183*x^7/7 + 319*x^8/8 + 433*x^9/9 + 762*x^10/10 + 1068*x^11/11 + 1625*x^12/12 +...
such that
exp(L(x)) = 1 + x + 4*x^2 + 10*x^3 + 26*x^4 + 59*x^5 + 140*x^6 + 307*x^7 + 684*x^8 + 1464*x^9 + 3122*x^10 + 6500*x^11 + 13426*x^12 +...+ A000293(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..72

Cf. A000293, A278403.

A143123 a(n) = Sum_{j=0..n} A000293(j).

Original entry on oeis.org

1, 2, 6, 16, 42, 101, 241, 548, 1232, 2696, 5818, 12318, 25744, 52992, 107796, 216598, 430669, 847518, 1652642, 3194279, 6124608, 11653341, 22015653, 41310879, 77024333, 142739427, 262996080, 481889660, 878308359, 1592707740

Offset: 0

Gary W. Adamson, Jul 26 2008

nonn

			a(3) = 16 = (1 + 1 + 4 + 10).

Cf. A000293.

a(11) corrected and more terms from Georg Fischer, Aug 28 2020

A305842 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A000293 (solid partitions).

Original entry on oeis.org

1, 4, 6, 14, 15, 26, 26, 48, 46, 83, 97, 146, 112, 49, -186, -448, -735, -485, 779, 3977, 9323, 16569, 23056, 23996, 10116, -31501, -120720, -283153, -548924, -932348, -1380125, -1655520, -1144651, 1384894, 7943203, 21083482, 42787785, 71816970, 98995196

Offset: 1

Ilya Gutkovskiy, Jun 11 2018

sign

Inverse weigh transform of A000293.

			(1 + x) * (1 + x^2)^4 * (1 + x^3)^6 * (1 + x^4)^14 * (1 + x^5)^15 * ... * (1 + x^n)^a(n) * ... = 1 + x + 4*x^2 + 10*x^3 + 26*x^4 + 59*x^5 + ... + A000293(k)*x^k + ...

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Solid Partition

Cf. A000293, A001511, A037452, A193718, A305841.

Product_{n>=1} (1 + x^n)^a(n) = Sum_{k>=0} A000293(k)*x^k.

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Original entry on oeis.org

1, 1, 4, 10, 26, 59, 141, 310, 692, 1483, 3162, 6583, 13602, 27613, 55579, 110445, 217554, 424148, 820294, 1572647, 2992892, 5652954, 10605608, 19765082, 36609945, 67405569, 123412204, 224728451, 407119735, 733878402, 1316631730, 2351322765, 4180714647, 7401898452, 13051476707, 22922301583, 40105025130, 69909106888, 121427077241, 210179991927, 362583131144

Offset: 0

N. J. A. Sloane

Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - Emeric Deutsch, Mar 23 2005
Euler transform of the triangular numbers 1,3,6,10,...
Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - Gary W. Adamson, Jun 13 2009
The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - Vaclav Kotesovec, May 12 2015]
Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - Alimzhan Amanov, Jul 13 2021

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
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.
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. [Annotated scanned copy]
R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605 [cond-mat.stat-mech], 2014.
Steven Finch, Integer Partitions, September 22, 2004, page 2. [Cached copy, with permission of the author]
Vaclav Kotesovec, Graph - The asymptotic ratio
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659; arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000293, A007294, A007326, A255939, A028377.
Cf. also A000041, A000219, A000335, A000391, A000417, A000428, A255965.
Cf. also A278403 (log of o.g.f.).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> n*(n+1)/2): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2008

a[0] = 1; a[n_] := a[n] = 1/(2*n)*Sum[(DivisorSigma[2, k]+DivisorSigma[3, k])*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)/2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 11 2015 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^3/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) = (1/(2*n))*Sum_{k=1..n} (sigma[2](k)+sigma[3](k))*a(n-k). - Vladeta Jovovic, Sep 17 2002
a(n) ~ Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4)*Pi^5) + 15^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2)*Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3*15^(1/4))) / (A^(1/2) * 2^(157/96) * 15^(13/96) * n^(61/96)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 11 2015
G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 21 2018

More terms from Sascha Kurz, Aug 15 2002

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)
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. [Annotated scanned copy]

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

cp's OEIS Frontend

A007326 Difference between A000294 and the number of solid partitions of n (A000293).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A037452 Let F(x) = 1 + 1*x + 4*x^2 + 10*x^3 + ..., the g.f. for A000293 (solid partitions), and write F(x) = 1/Product_{n>=1} (1 - x^n)^a(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A002836 Let F(x) = 1 + x + 4x^2 + 10x^3 + ... = g.f. for A000293 (solid partitions) and expand (1-x)(1-x^2)(1-x^3)...*F(x) in powers of x.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A161564 Convolution triangle for A000293 (solid partitions): A000041 * A002836.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A193718 Let F(x) = 1 + 1*x + 4*x^2 + 10*x^3 + ..., the g.f. for A000293 (solid partitions), and write F(x) = 1/Product_{n>=1} E(n)^a(n) where E(n) = Product_{k>=n} (1 - x^(n*k)).

Views

Author

Keywords

Crossrefs

A277613 Logarithmic derivative of the g.f. of the solid partitions A000293.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A143123 a(n) = Sum_{j=0..n} A000293(j).

Views

Author

Keywords

Examples

Crossrefs

Extensions

A305842 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A000293 (solid partitions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Views

A037452 Let F(x) = 1 + 1x + 4x^2 + 10*x^3 + ..., the g.f. for A000293 (solid partitions), and write F(x) = 1/Product_{n>=1} (1 - x^n)^a(n).

A193718 Let F(x) = 1 + 1x + 4x^2 + 10x^3 + ..., the g.f. for A000293 (solid partitions), and write F(x) = 1/Product_{n>=1} E(n)^a(n) where E(n) = Product_{k>=n} (1 - x^(nk)).