A000219 -id:A000219 - cp's OEIS frontend

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518

Offset: 0

N. J. A. Sloane

An ordinary partition is a row of numbers in nondecreasing order whose sum is n. Here the numbers are in a three-dimensional pile, nondecreasing in the x-, y- and z-directions.
Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.
Equals A000041 convolved with A002836: [1, 0, 2, 5, 12, 24, 56, 113, ...] and row sums of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

			Examples for n=2 and n=3.
a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.
a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.
From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(4) = 26 solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((21))           ((22))
         ((1)(1))    ((111))          ((31))
         ((1))((1))  ((2)(1))         ((211))
                     ((11)(1))        ((1111))
                     ((2))((1))       ((2)(2))
                     ((1)(1)(1))      ((3)(1))
                     ((11))((1))      ((21)(1))
                     ((1)(1))((1))    ((11)(11))
                     ((1))((1))((1))  ((111)(1))
                                      ((2))((2))
                                      ((3))((1))
                                      ((2)(1)(1))
                                      ((21))((1))
                                      ((11))((11))
                                      ((11)(1)(1))
                                      ((111))((1))
                                      ((2)(1))((1))
                                      ((1)(1)(1)(1))
                                      ((11)(1))((1))
                                      ((2))((1))((1))
                                      ((1)(1))((1)(1))
                                      ((1)(1)(1))((1))
                                      ((11))((1))((1))
                                      ((1)(1))((1))((1))
                                      ((1))((1))((1))((1))
(End)

P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
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
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. [Annotated scanned copy], DOI
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666.
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, arXiv:1406.5605 [cond-mat.stat-mech], 2014; J. Stat. Phys. 158 (2015) 950-967.
Suresh Govindarajan, Solid Partitions Project Dec 14, 2010.
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
P. A. MacMahon, Combinatory analysis.
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003; J. Phys. A 36 (2003), no. 24, 6651-6659.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
Eric Weisstein's World of Mathematics, Solid Partition
Wikipedia, Solid partition
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A007326, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim).
Cf. A161564. - Gary W. Adamson, Jun 13 2009
Cf. A001970, A002974, A003293, A007713, A114736, A117433, A323657.

planePtns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And@@(GreaterEqual@@@Transpose[PadRight[#]])&],{ptn,IntegerPartitions[n]}];
solidPtns[n_]:=Join@@Table[Select[Tuples[planePtns/@y],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])&],{y,IntegerPartitions[n]}];
Table[Length[solidPtns[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from the Mustonen and Rajesh article, May 02 2003
a(51)-a(62) found by Suresh Govindarajan and students, Dec 14 2010
a(63)-a(68) found by Suresh Govindarajan and students, Jun 01 2011
a(69)-a(72) found by Suresh Govindarajan and Srivatsan Balakrishnan, Jan 03 2013

A004250 Number of partitions of n into 3 or more parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, 127, 168, 222, 288, 375, 480, 616, 781, 990, 1243, 1562, 1945, 2422, 2996, 3703, 4550, 5588, 6826, 8332, 10126, 12292, 14865, 17958, 21618, 25995, 31165, 37317, 44562

Offset: 1

N. J. A. Sloane

Number of (n+1)-vertex spider graphs: trees with n+1 vertices and exactly 1 vertex of degree at least 3 (i.e. branching vertex). There is a trivial bijection with the objects described in the definition. - Emeric Deutsch, Feb 22 2014

Also the number of graphical partitions of 2n into n parts. - Gus Wiseman, Jan 08 2021

			a(6)=7 because there are three partitions of n=6 with i=3 parts: [4, 1, 1], [3, 2, 1], [2, 2, 2] and two partitions with i=4 parts: [3, 1, 1, 1], [2, 2, 1, 1] and one partition with i=5 parts: [2, 1, 1, 1, 1] and one partition with i=6 parts: [1, 1, 1, 1, 1, 1].
From _Gus Wiseman_, Jan 18 2021: (Start)
The a(3) = 1 through a(7) = 11 graphical partitions of 2n into n parts:
  (222)  (2222)  (22222)  (222222)  (2222222)
         (3221)  (32221)  (322221)  (3222221)
                 (33211)  (332211)  (3322211)
                 (42211)  (333111)  (3332111)
                          (422211)  (4222211)
                          (432111)  (4322111)
                          (522111)  (4331111)
                                    (4421111)
                                    (5222111)
                                    (5321111)
                                    (6221111)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
N. Metropolis and P. R. Stein, The enumeration of graphical partitions, Europ. J. Combin., 1 (1980), 139-1532.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Eric Weisstein's World of Mathematics. Spider Graph
Wikipedia, Starlike tree
Index entries for sequences related to graphical partitions

Cf. A004251, A029889, A035300, A095268, A058984.
Rightmost column of A259873.
Central column of A339659.
A000041 counts partitions of 2n into n parts, ranked by A340387.
A000569 counts graphical partitions, ranked by A320922.
A008284 counts partitions by sum and length.
A027187 counts partitions of even length.
A309356 ranks simple covering graphs.
The following count vertex-degree partitions and give their Heinz numbers:
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339617 counts non-graphical partitions of 2n (A339618).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A000219, A339560, A339561, A339661.
Partial sums of A117995.

with(combinat);
for i from 1 to 15 do pik(i,3) od;
pik:= proc(n::integer, k::integer)
# Thomas Wieder, Jan 30 2007
local i, Liste, Result;
if k > n or n < 0 or k < 1 then
return fail
end if;
Result := 0;
for i from k to n do
Liste:= PartitionList(n,i);
#print(Liste);
Result := Result + nops(Liste);
end do;
return Result;
end proc;
PartitionList := proc (n, k)
# Authors: Herbert S. Wilf and Joanna Nordlicht. Source: Lecture Notes
# "East Side West Side,..." University of Pennsylvania, USA, 2002.
# Available at: http://www.cis.upenn.edu/~wilf/lecnotes.html
# Calculates the partition of n into k parts.
# E.g. PartitionList(5,2) --> [[4, 1], [3, 2]].
local East, West;
if n < 1 or k < 1 or n < k then
RETURN([])
elif n = 1 then
RETURN([[1]])
else if n < 2 or k < 2 or n < k then
West := []
else
West := map(proc (x) options operator, arrow;
[op(x), 1] end proc,PartitionList(n-1,k-1)) end if;
if k <= n-k then
East := map(proc (y) options operator, arrow;
map(proc (x) options operator, arrow; x+1 end proc,y) end proc,PartitionList(n-k,k))
else East := [] end if;
RETURN([op(West), op(East)])
end if;
end proc;
#  Thomas Wieder, Feb 01 2007
ZL :=[S, {S = Set(Cycle(Z),3 <= card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..41); # Zerinvary Lajos, Mar 25 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card >=3)},unlabelled]: seq(combstruct[count](B, size=n), n=1..41); # Zerinvary Lajos, Mar 21 2009

Length /@ Table[Select[Partitions[n], Length[#] > 2 &], {n, 20}] (* Eric W. Weisstein, May 16 2007 *)
Table[Count[Length /@ Partitions[n], ?(# > 2 &)], {n, 20}] (* _Eric W. Weisstein, May 16 2017 *)
Table[PartitionsP[n] - Floor[n/2] - 1, {n, 20}] (* Eric W. Weisstein, May 16 2017 *)
Length /@ Table[IntegerPartitions[n, {3, n}], {n, 20}] (* Eric W. Weisstein, May 16 2017 *)

a(n) = numbpart(n) - (n+2)\2; /* Joerg Arndt, Apr 03 2013 */

G.f.: Sum_{n>=0} (q^n / Product_{k=1..n+3} (1 - q^k)). -  N. J. A. Sloane
a(n) = A000041(n) - floor((n+2)/2) = A000041(n)-A004526(n+2) = A058984(n)-1. - Vladeta Jovovic, Jun 18 2003
Let P(n,i) denote the number of partitions of n into i parts. Then a(n) = Sum_{i=3..n} P(n,i). - Thomas Wieder, Feb 01 2007
a(n) = A259873(n,n). - Gus Wiseman, Jan 08 2021

Definition corrected by Thomas Wieder, Feb 01 2007 and by Eric W. Weisstein, May 16 2007

A005380 Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).

Original entry on oeis.org

1, 2, 6, 14, 33, 70, 149, 298, 591, 1132, 2139, 3948, 7199, 12894, 22836, 39894, 68982, 117948, 199852, 335426, 558429, 922112, 1511610, 2460208, 3977963, 6390942, 10206862, 16207444, 25596941, 40214896, 62868772, 97814358

Offset: 0

N. J. A. Sloane

Also, a(n) = number of partitions of the integer n where there are k+1 different kinds of part k for k = 1, 2, 3, ....
Also, a(n) = number of partitions of n objects of 2 colors. These are set partitions, the n objects are not labeled but colored, using two colors.  For each subset of size k there are k+1 different possibilities, i=0..k white and k-i black objects.
Also, a(n) = number of simple unlabeled graphs with n nodes of 2 colors whose components are complete graphs. - Geoffrey Critzer, Sep 27 2012

			We represent each summand, k, in a partition of n as k identical objects. Then we color each object. We have no regard for the order of the colored objects.
a(3) = 14 because we have:  www; wwb; wbb; bbb; ww + w; ww + b;  wb + w; wb + b; bb + w; bb + b; w + w + w; w + w + b; w + b + b; b + b + b, where the 2 colors are black b and white w. - _Geoffrey Critzer_, Sep 27 2012
a(3) = 14 because we have:  3; 3'; 3''; 3'''; 2 + 1; 2 + 1';  2' + 1; 2' + 1'; 2'' + 1; 2'' + 1'; 1 + 1 + 1; 1 + 1 + 1'; 1 + 1' + 1'; 1' + 1' + 1', where a part k of different sorts is given as k, k', k'', etc. - _Joerg Arndt_, Mar 09 2015
From _Alois P. Heinz_, Mar 09 2015: (Start)
The a(4) = 33 = 5 + 9 + 6 + 8 + 5 partitions of 4 objects of 2 colors are:
5 partitions for the integer partition of 4 = 1 + 1 + 1 + 1:
  01: {{b}, {b}, {b}, {b}}
  02: {{b}, {b}, {b}, {w}}
  03: {{b}, {b}, {w}, {w}}
  04: {{b}, {w}, {w}, {w}}
  05: {{w}, {w}, {w}, {w}}
9 partitions for the integer partition of 4 = 1 + 1 + 2:
  06: {{b}, {b}, {b,b}}
  07: {{b}, {w}, {b,b}}
  08: {{w}, {w}, {b,b}}
  09: {{b}, {b}, {w,b}}
  10: {{b}, {w}, {w,b}}
  11: {{w}, {w}, {w,b}}
  12: {{b}, {b}, {w,w}}
  13: {{b}, {w}, {w,w}}
  14: {{w}, {w}, {w,w}}
6 partitions for the integer partition of 4 = 2 + 2:
  15: {{b,b}, {b,b}}
  16: {{b,b}, {w,b}}
  17: {{b,b}, {w,w}}
  18: {{w,b}, {w,b}}
  19: {{w,b}, {w,w}}
  20: {{w,w}, {w,w}}
8 partitions for the integer partition of 4 = 1 + 3:
  21: {{b}, {b,b,b}}
  22: {{w}, {b,b,b}}
  23: {{b}, {w,b,b}}
  24: {{w}, {w,b,b}}
  25: {{b}, {w,w,b}}
  26: {{w}, {w,w,b}}
  27: {{b}, {w,w,w}}
  28: {{w}, {w,w,w}}
5 partitions for the integer partition of 4 = 4:
  29: {{b,b,b,b}}
  30: {{w,b,b,b}}
  31: {{w,w,b,b}}
  32: {{w,w,w,b}}
  33: {{w,w,w,w}}
Some see number partitions, others see set partitions, ...
(End)
It is obvious from the example of _Alois P. Heinz_ that a(n) enumerates multi-set partitions of a multi-set of n elements of two kinds. In the case that there is only one kind, this reduces to the usual case of numerical partitions. In the case that all the n elements are distinct, then this reduces to the case of set partitions. - _Michael Somos_, Mar 09 2015
There are a(3) = 14 plane partitions of 6 with trace 3; of 7 with trace 4; of 8 with trace 5; etc. See a formula above with the Stanley Exercise 7.99. - _Wolfdieter Lang_, Mar 09 2015
From _Daniel Forgues_, Mar 09 2015: (Start)
The a(3) = 14 = 4 + 6 + 4 partitions of 3 objects of 2 colors are:
4 partitions for the integer partition of 3 = 1 + 1 + 1:
  01: {{b}, {b}, {b}}
  02: {{b}, {b}, {w}}
  03: {{b}, {w}, {w}}
  04: {{w}, {w}, {w}}
6 partitions for the integer partition of 3 = 1 + 2:
  05: {{b}, {b,b}}
  06: {{w}, {b,b}}
  07: {{b}, {w,b}}
  08: {{w}, {w,b}}
  09: {{b}, {w,w}}
  10: {{w}, {w,w}}
4 partitions for the integer partition of 3 = 3:
  11: {{b,b,b}}
  12: {{w,b,b}}
  13: {{w,w,b}}
  14: {{w,w,w}}
The a(2) = 6 = 3 + 3 partitions of 2 objects of 2 colors are:
3 partitions for the integer partition of 2 = 1 + 1:
  01: {{b}, {b}}
  02: {{b}, {w}}
  03: {{w}, {w}}
3 partitions for the integer partition of 2 = 2:
  04: {{b,b}}
  05: {{w,b}}
  06: {{w,w}}
The a(1) = 2 partitions of 1 object of 2 colors are:
2 partitions for the integer partition of 1 = 1:
  01: {{b}}
  02: {{w}}
a(0) = 1: the empty partition, since empty sum is 0.
Triangle(sort of, since n_th row has p(n) = A000041 terms):
  1:  2
  2:  3, 3
  3:  4, 6, 4
  4:  5, 9, 6, 8, 5
  5:  6, ?, ?, ?, ?, ?, 6
  6:  7, ?, ?, ?, ?, ?, ?, ?, ?, ?, 7
Can we find a recurrence relation? (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 7.99, p. 484 and pp. 548-549.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Carlos A. A. Florentino, Plethystic exponential calculus and permutation polynomials, arXiv:2105.13049 [math.CO], 2021. Mentions this sequence.
Vaclav Kotesovec, Graph - The asymptotic ratio
P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
N. J. A. Sloane, Transforms
R. P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279.
R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combin. Theory, vol. A14 53-65 1973, esp. p. 64.

Row sums of A054225.
Column k=2 of A075196.
Cf. A000219, A219555, A217093, A255050, A255052, A089353, A298988.

mul( (1-x^i)^(-i-1),i=1..80); series(%,x,80); seriestolist(%);
# second Maple program:
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+1): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

max = 31; f[x_] = Product[ 1/(1-x^k)^(k+1), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 08 2011, after g.f. *)
etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n==0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; a = etr[#+1&]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

a(n)=polcoeff(prod(i=1,n,(1-x^i+x*O(x^n))^-(i+1)),n)

EULER transform of b(n) = n+1.
a(n) ~ Zeta(3)^(13/36) * exp(1/12 - Pi^4/(432*Zeta(3)) + Pi^2 * n^(1/3) / (3*2^(4/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 2^(23/36) * 3^(1/2) * Pi * n^(31/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 07 2015
a(n) = A089353(n+m, m), n >= 1, for each m >= n. a(0) =1. See the Stanley reference, Exercise 7.99. - Wolfdieter Lang, Mar 09 2015
G.f.: exp(Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Aug 11 2018

Edited by Christian G. Bower, Sep 07 2002
New name from Joerg Arndt, Mar 09 2015
Restored 1995 name. - N. J. A. Sloane, Mar 09 2015

A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 17, 28, 41, 75, 122, 192, 314, 484, 771, 1216, 1861, 2848, 4395, 6610, 10037

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
From Gus Wiseman, Jan 17 2019: (Start)
Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:
  6   51   42   321
.
  5   4   41   31   32   31   22   221   211
  1   2   1    2    1    11   2    1     11
.
  3   21   21   111
  2   2    11   11
  1   1    1    1
(End)

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{2},{1,2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{1,1},{1,1,1}}
   {{1,2},{1,2,2}}
   {{2,2},{1,2,2}}

Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646.

Cf. A000085, A138178, A323436.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,UnsameQ@@Transpose[PadRight[#]],And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}] (* Gus Wiseman, Jan 18 2019 *)

a(11)-a(17) from Gus Wiseman, Jan 18 2019

a(18)-a(21) from Robert Price, Jun 21 2021

A073592 Euler transform of negative integers.

Original entry on oeis.org

1, -1, -2, -1, 0, 4, 4, 7, 3, -2, -9, -17, -25, -24, -13, -1, 32, 61, 97, 111, 112, 74, 8, -108, -243, -392, -512, -569, -542, -358, -33, 473, 1078, 1788, 2395, 2865, 2955, 2569, 1496, -245, -2751, -5783, -9121, -12299, -14739, -15806, -14719, -10930, -3813, 6593, 20284, 36139, 53081, 68620, 80539

Offset: 0

Vladeta Jovovic, Aug 28 2002

sign

1/A(x) is g.f. for A000219.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
E. M. Wright, Coefficients of a reciprocal generating function, Quart. J. Math. 17 (1) (1966) 39-43, ADS Abstracts.
N. J. A. Sloane, Transforms

Column k=1 of A283272.

Cf. A000219, A001157, A001478, A026007, A156616, A255528.

a:= proc(n) option remember; `if`(n=0, 1, -add(
      numtheory[sigma][2](j)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 12 2015

nmax=50; CoefficientList[Series[Exp[Sum[-x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 02 2015 *)
a[n_]:= a[n] = -1/n*Sum[DivisorSigma[2,k]*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 02 2015 *)

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n)
print([b(n) for n in range(55)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k>0} (1-x^k)^k.
a(n) = -1/n*Sum_{k=1..n} sigma[2](k)*a(n-k).
G.f.: exp( Sum_{n>=1} -sigma_2(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017

A000335 Euler transform of A000292.

Original entry on oeis.org

1, 5, 15, 45, 120, 331, 855, 2214, 5545, 13741, 33362, 80091, 189339, 442799, 1023192, 2340904, 5302061, 11902618, 26488454, 58479965, 128120214, 278680698, 602009786, 1292027222, 2755684669, 5842618668, 12317175320, 25825429276, 53865355154, 111786084504, 230867856903, 474585792077, 971209629993

Offset: 1

N. J. A. Sloane

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=1..500
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]
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011, p. 20.
N. J. A. Sloane, Transforms

Cf. A000041, A000219, A000294, A000391, A000417, A000428, A255965.

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-> binomial(n+2,3)): seq(a(n), n=1..26); # Alois P. Heinz, Sep 08 2008

max = 33; f[x_] := Exp[ Sum[ x^k/(1-x^k)^4/k, {k, 1, max}]]; Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x], 1](* Jean-François Alcover, Nov 21 2011, after Joerg Arndt *)
nmax=50; Rest[CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]] (* Vaclav Kotesovec, Mar 11 2015 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, #*p[#] &]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Binomial[#+2, 3]&]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *)

a(n)=if(n<1, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^4/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

N=66; x='x+O('x^66); gf=-1 + exp(sum(k=1, N, x^k/(1-x^k)^4/k)); Vec(gf) /* Joerg Arndt, Jul 06 2011 */

# uses[EulerTransform from A166861] and prepends a(0) = 1.
a = EulerTransform(lambda n: n*(n+1)*(n+2)//6)
print([a(n) for n in range(33)]) # Peter Luschny, Nov 17 2022

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 - Zeta(3) / (8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8*Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2 / (90*Zeta(5)) + Zeta'(-3)/6 + (Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5)) - Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5))) * n^(2/5) + Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - Vaclav Kotesovec, Mar 12 2015

A358914 Number of twice-partitions of n into distinct strict partitions.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 13, 20, 32, 51, 83, 130, 206, 320, 496, 759, 1171, 1786, 2714, 4104, 6193, 9286, 13920, 20737, 30865, 45721, 67632, 99683, 146604, 214865, 314782, 459136, 668867, 972425, 1410458, 2040894, 2950839, 4253713, 6123836, 8801349, 12627079

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(6) = 13 twice-partitions:
  ((1))  ((2))  ((3))     ((4))      ((5))      ((6))
                ((21))    ((31))     ((32))     ((42))
                ((2)(1))  ((3)(1))   ((41))     ((51))
                          ((21)(1))  ((3)(2))   ((321))
                                     ((4)(1))   ((4)(2))
                                     ((21)(2))  ((5)(1))
                                     ((31)(1))  ((21)(3))
                                                ((31)(2))
                                                ((3)(21))
                                                ((32)(1))
                                                ((41)(1))
                                                ((3)(2)(1))
                                                ((21)(2)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..100

The unordered version is A050342, non-strict A261049.
This is the distinct case of A270995.
The case of strictly decreasing sums is A279785.
The case of constant sums is A279791.
For distinct instead of weakly decreasing sums we have A336343.
This is the twice-partition case of A358913.
A001970 counts multiset partitions of integer partitions.
A055887 counts sequences of partitions.
A063834 counts twice-partitions.
A330462 counts set systems by total sum and length.
A358830 counts twice-partitions with distinct lengths.
Cf. A000009, A000219, A075900, A271619, A296122, A304969, A321449, A336342, A358901, A358906, A358907.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]

seq(n,k)={my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))-1)); Vec(prod(k=1, n, my(c=u[k]); sum(j=0, min(c,n\k), x^(j*k)*c!/(c-j)!,  O(x*x^n))))} \\ Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A003293 Number of planar partitions of n decreasing across rows.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 56, 90, 143, 223, 348, 532, 811, 1224, 1834, 2725, 4031, 5914, 8638, 12540, 18116, 26035, 37262, 53070, 75292, 106377, 149738, 209980, 293473, 408734, 567484, 785409, 1083817, 1491247, 2046233, 2800125, 3821959, 5203515

Offset: 0

N. J. A. Sloane

Also number of planar partitions monotonically decreasing down antidiagonals (i.e., with b(n,k) <= b(n-1,k+1)). Transpose (to get planar partitions decreasing down columns), then take the conjugate of each row. - Franklin T. Adams-Watters, May 15 2006
Also number of partitions into one kind of 1's and 2's, two kinds of 3's and 4's, three kinds of 5's and 6's, etc. - Joerg Arndt, May 01 2013
Also count of semistandard Young tableaux with sum of entries equal to n (row sums of A228125). - Wouter Meeussen, Aug 11 2013

			From _Gus Wiseman_, Jan 17 2019: (Start)
The a(6) = 21 plane partitions with strictly decreasing columns (the count is the same as for strictly decreasing rows):
  6   51   42   411   33   321   3111   222   2211   21111   111111
.
  5   4   41   31   32   311   22   221   2111
  1   2   1    2    1    1     11   1     1
.
  3
  2
  1
(End)

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 133.
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..10000 (first 1001 terms from Nathaniel Johnston)
M. S. Cheema and W. E. Conway, Numerical investigation of certain asymptotic results in the theory of partitions, Math. Comp., 26 (1972), 999-1005.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
B. Gordon and L. Houten, Notes on Plane Partitions I, J. of Comb. Theory, 4 (1968), 72-80.
B. Gordon and L. Houten, Notes on Plane Partitions II, J. of Comb. Theory, 4 (1968), 81-99.
Basil Gordon and Lorne Houten, Notes on plane partitions III (first page is available), Duke Math. J. Volume 36, Number 4 (1969), 801-824.
B. Gordon and L. Houten, Notes on Plane Partitions V, Journal of Combinatorial Theory, vol. 11, issue 2, 1971, pp. 157-168.
B. Gordon and L. Houten, Notes on Plane Partitions VI, Discrete Mathematics, vol. 26, issue 1, 1979, pp. 41-45.
Vaclav Kotesovec, Graph - asymptotic ratio for 10000 terms.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Richard P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279.
Richard P. Stanley, Theory and Application of Plane Partitions. Part 2, Studies in Appl. Math., 1 (1971), 259-279.

Cf. A005308, A005986.

Cf. A000085, A000219, A053529, A138178, A323432, A323436.

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-> `if`(modp(n,2)=0,n,n+1)/2): seq(a(n), n=0..45);  # Alois P. Heinz, Sep 08 2008

CoefficientList[Series[Product[(1-x^k)^(-Ceiling[k/2]), {k, 1, 40}], {x, 0, 40}], x][[1 ;; 40]] (* Jean-François Alcover, Apr 18 2011, after Michael Somos *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^((2*k+1-(-1)^k)/4),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)
nmax = 50; CoefficientList[Series[Product[1/((1-x^(2*k-1))*(1-x^(2*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2015 *)

{a(n)=if(n<0, 0, polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n))} /* Michael Somos, Sep 19 2006 */

G.f.: Product_(1 - x^k)^{-c(k)}, c(k) = 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ....
Euler transform of A110654. - Michael Somos, Sep 19 2006
a(n) ~ 2^(-3/4) * (3*Pi*Zeta(3))^(-1/2) * (n/Zeta(3))^(-49/72) * exp(3/2*Zeta(3) * (n/Zeta(3))^(2/3) + Pi^2*(n/Zeta(3))^(1/3)/24 - Pi^4/(3456*Zeta(3)) + Zeta'(-1)/2) [Basil Gordon and Lorne Houten, 1969]. - Vaclav Kotesovec, Feb 28 2015

More terms from James Sellers, Feb 06 2000

Additional comments from Michael Somos, May 19 2000

A218482 First differences of the binomial transform of the partition numbers (A000041).

Original entry on oeis.org

1, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812, 309274516740

Offset: 0

Paul D. Hanna, Oct 29 2012

nonn

a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.
From Gus Wiseman, Dec 12 2022: (Start)
Also the number of sequences of compositions (A133494) with weakly decreasing lengths and total sum n. For example, the a(0) = 1 through a(3) = 8 sequences are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((12))
             ((1)(1))  ((21))
                       ((111))
                       ((1)(2))
                       ((2)(1))
                       ((11)(1))
                       ((1)(1)(1))
The case of constant lengths is A101509.
The case of strictly decreasing lengths is A129519.
The case of sequences of partitions is A141199.
The case of twice-partitions is A358831.
(End)

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
The g.f. equals the product:
A(x) = (1-x)/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1  +  x*(1-x)/((1-x)-x)^2  +  x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2  +  x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2  +  x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000219, A011782, A055887, A063834, A075900, A098407, A101509, A103446, A129519, A141199, A218481.

b:= proc(n) option remember;
      add(combinat[numbpart](k)*binomial(n,k), k=0..n)
    end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 19 2014

Flatten[{1, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

{a(n)=sum(k=0,n,(binomial(n,k)-if(n>0,binomial(n-1,k)))*numbpart(k))}
for(n=0,40,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1-x)^k/((1-x)^k-X^k)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,x^m*(1-x)^(m*(m-1)/2)/prod(k=1,m,((1-x)^k - X^k))),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,x^(m^2)*(1-X)^m/prod(k=1,m,((1-x)^k - x^k)^2)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(exp(sum(m=1,n+1,x^m/((1-x)^m-X^m)/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(exp(sum(m=1,n+1,sigma(m)*x^m/(1-X)^m/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1 + x^k/(1-X)^k)^valuation(2*k,2)),n)}

G.f.: Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
G.f.: Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
G.f.: Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
G.f.: exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
G.f.: exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).
G.f.: Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

cp's OEIS Frontend

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

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A004250 Number of partitions of n into 3 or more parts.

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A005380 Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).

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A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

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A073592 Euler transform of negative integers.

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A000335 Euler transform of A000292.

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A358914 Number of twice-partitions of n into distinct strict partitions.