A000219 -id:A000219 - cp's OEIS frontend

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
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).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
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.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

A006171 Number of factorization patterns of polynomials of degree n over integers.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 34, 52, 94, 145, 244, 370, 603, 899, 1410, 2087, 3186, 4650, 6959, 10040, 14750, 21077, 30479, 43120, 61574, 86308, 121785, 169336, 236475, 326201, 451402, 618135, 848209, 1153733, 1571063, 2123325, 2871419, 3857569, 5182999, 6924303

Offset: 0

N. J. A. Sloane

Number of partitions of n where there are unlimited distinguishable but unlabeled objects of each size. E.g., in splitting 2 into two parts of size 1, we distinguish whether the same object is used for each part. Also number of factorization patterns over rationals, or many other UFDs (but not over real or complex numbers). - Franklin T. Adams-Watters, Jun 19 2006
Equals the "aerate and convolve" convergent of A000041: (1, 1, 2, 3, 5, 7, 11, ...) * (1, 0, 1, 0, 2, 0, 3, 0, 5, ...) * (1, 0, 0, 1, 0, 0, 2, 0, 0, 3, ...). - Gary W. Adamson, Jun 16 2009
Also equals the number of distinct (up to unitary similarity) unital *-subalgebras of the n X n complex matrices. A unital *-subalgebra is a subspace that is closed under multiplication and the conjugate transpose, and which contains the identity matrix (see A215905 and A215925). - Nathaniel Johnston, Aug 27 2012
Also equals the number of partitions having parts consisting of runs of equal parts. - Gregory L. Simay, May 25 2017
Also equals the number of generalized partitions of n when there are d(a) different types of a, (a = 1,2,3,...), where d(n) is the number of divisors of n. a(3)=5 because there are 5 partitions of 3 with "d(a) copies of a", namely (3_1), (3_2), (2_1, 1_1), (2_2, 1_1), (1_1, 1_1, 1_1). - Augustine O. Munagi, Jun 13 2022

			For n=3 we have 3 = (3*1) = (1*3) = (2*1) + (1*1) = (1*2) + (1*1) = (1*1) + (1*1) + (1*1) so a(3)=5.
For n=4 we have the following 11 partitions, with the additive runs indicated by "[]": [4], [3]+[1], [2+2], [2]+[2], [2]+[1+1], [2]+[1]+[1], [1+1+1+1], [1+1+1]+[1], [1+1]+[1+1], [1+1]+[1]+[1], [1]+[1]+[1]+[1]. - _Gregory L. Simay_, May 25 2017

R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82.
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 T. D. Noe)
A. K. Agarwal and G. L. Mullen, Partitions with "d(a) copies of a", J. Combin. Theory Ser. A, 48(1)(1988), 120-135.
N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), pp. 182-191.
William Q. Erickson, The Demazure product extended to biwords, arXiv:2407.13165 [math.CO], 2024. See p. 16.
MathOverflow, Number of representations of an integer as an (arbitrary) sum of products, 2014.
N. J. A. Sloane, Transforms

Cf. A000005, A060640, A061255, A061256, A001970, A061257.
Cf. A006167-A006170, A006171 (log).
Cf. A000041, A115621, A215905, A215909, A215914, A215925, A288098.
Cf. A000219.

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(tau): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

max = 50; gf[x_] := Product[(1 - x^k)^-DivisorSigma[0, k], {k, 1, max}]; CoefficientList[ Series[gf[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 23 2011 *)
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2018, the fastest *)
nmax = 50; CoefficientList[Series[Product[Sum[PartitionsP[k]*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 26 2020 *)

{a(n) = if(n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x * O(x^n))^numdiv(k)), n))}; /* Michael Somos, Apr 01 2003 */

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)); Vec(gf) \\ Joerg Arndt, May 03 2008

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

{A060640(n)=sumdiv(n, d, d*sigma(n/d))}
{a(n)=polcoeff(exp(sum(m=1,n+1,A060640(m)*x^m/m)+x*O(x^n)),n)} /* Paul D. Hanna, Oct 19 2011 */

From Vladeta Jovovic, Apr 21 2001: (Start)
Euler transform of tau(n), tau(n) = the number of divisors of n, cf. A000005.
G.f.: Product_{k>=1} (1 - x^k)^(-tau(k)).
a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k) = Sum_{d|k} d*tau(d), cf. A060640. (End)
a(n) = Sum_{ partition of n} product p(k(i)), where p(n) is the partition function A000041. E.g., for the partition [4,2^3,1^4], the product is p(1)*p(3)*p(4) = 1*3*5 = 15. - Franklin T. Adams-Watters, Jun 19 2006
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)/n ). - Paul D. Hanna, Mar 28 2009
From Paul D. Hanna, Oct 19 2011: (Start)
Logarithmic derivative yields A060640.
G.f.: A(x) = exp( Sum_{n>=1} A060640(n)*x^n/n ), where A060640(n) = Sum_{d|n} d*sigma(n/d). (End)
G.f.: 1/Product_{n>=1} E(q^n) where E(q) = Product_{n>=1} (1-q^n). - Joerg Arndt, Feb 27 2014
log(a(n)) ~ Pi * sqrt(n*log(n)/3) [Brigham, 1950]. - Vaclav Kotesovec, Jan 04 2017
a(n) ~ exp(Pi*sqrt(n/(3*log(n))) * (log(n) - log(log(n))/2 + gamma + 6*Zeta'(2)/Pi^2 + log(2/Pi) + log(3)/2)) * Pi^(1/4) * (log(n))^(1/8) / (2^(3/4) * 3^(1/8) * n^(5/8)), where gamma is the Euler-Mascheroni constant (A001620) and Zeta'(2) = -0.9375482543158437537... (see A073002) [user Lucia, MathOverflow, 2014]. - Vaclav Kotesovec, Jan 05 2017

A319616 Number of non-isomorphic square multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 80, 230, 719, 2271, 7519, 25425, 88868, 317972, 1168360, 4392724, 16903393, 66463148, 266897917, 1093550522, 4568688612, 19448642187, 84308851083, 371950915996, 1669146381915, 7615141902820, 35304535554923, 166248356878549, 794832704948402, 3856672543264073, 18984761300310500

Offset: 0

Gus Wiseman, Sep 25 2018

nonn

A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices.

Also the number of square integer matrices with entries summing to n and no empty rows or columns, up to permutation of rows and columns.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1}, {2}}
3: {{1,1,1}}
   {{1}, {2,2}}
   {{2}, {1,2}}
   {{1}, {2},{3}}
4: {{1,1,1,1}}
   {{1}, {1,2,2}}
   {{1}, {2,2,2}}
   {{2}, {1,2,2}}
   {{1,1}, {2,2}}
   {{1,2}, {1,2}}
   {{1,2}, {2,2}}
   {{1}, {1}, {2,3}}
   {{1}, {2}, {3,3}}
   {{1}, {3}, {2,3}}
   {{1}, {2}, {3}, {4}}
Non-isomorphic representatives of the a(4) = 11 square matrices:
. [4]
.
. [1 0]   [1 0]   [0 1]   [2 0]   [1 1]   [1 1]
. [1 2]   [0 3]   [1 2]   [0 2]   [1 1]   [0 2]
.
. [1 0 0]   [1 0 0]   [1 0 0]
. [1 0 0]   [0 1 0]   [0 0 1]
. [0 1 1]   [0 0 2]   [0 1 1]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A321615.

Cf. A000219, A007716, A007718, A056156, A059201, A316980, A316983, A318795, A319560, A319616-A319646, A300913.

(* See A318795 for M[m, n, k]. *)
T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];
a[0] = 1; a[n_] := Sum[T[n, k], {k, 1, n}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 16}] (* Jean-François Alcover, Nov 24 2018, after Andrew Howroyd *)

\\ See A318795 for M.
a(n) = {if(n==0, 1, sum(i=1, n, M(i,i,n) - 2*M(i,i-1,n) + M(i-1,i-1,n)))} \\ Andrew Howroyd, Nov 15 2018

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n,polcoef(G(k,n,n,y),k,y) - polcoef(G(k-1,n,n,y),k,y)))} \\ Andrew Howroyd, Jan 15 2024

a(11)-a(20) from Andrew Howroyd, Nov 15 2018

a(21) onwards from Andrew Howroyd, Jan 15 2024

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

Original entry on oeis.org

1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699

Offset: 0

N. J. A. Sloane

In general, for t > 0, if g.f. = Product_{m>=1} (1 + t*q^m)^m then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (t+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(t) + log(t)^3 - 6*polylog(3, -1/t). - Vaclav Kotesovec, Jan 04 2016

			For n = 4, we have 8 partitions
  01: [4]
  02: [4']
  03: [4'']
  04: [4''']
  05: [3, 1]
  06: [3', 1]
  07: [3'', 1]
  08: [2, 2']

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
Vaclav Kotesovec, Graph - The asymptotic ratio
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.

Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857, A285223, A304040.
Cf. A000219. - Gary W. Adamson, Jun 13 2009
Cf. A027998, A248882, A248883, A248884.
Cf. A026011, A027346, A027906.
Column k=1 of A284992.

with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 03 2013

a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Apr 17 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

N=66; q='q+O('q^N);
gf= prod(n=1,N, (1+q^n)^n );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

a(n) = (1/n)*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
G.f.: Product_{m>=1} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006
a(n) ~ zeta(3)^(1/6) * exp((3/2)^(4/3) * zeta(3)^(1/3) * n^(2/3)) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where zeta(3) = A002117. - Vaclav Kotesovec, Mar 05 2015

A000569 Number of graphical partitions of 2n.

Original entry on oeis.org

1, 2, 5, 9, 17, 31, 54, 90, 151, 244, 387, 607, 933, 1420, 2136, 3173, 4657, 6799, 9803, 14048, 19956, 28179, 39467, 54996, 76104, 104802, 143481, 195485, 264941, 357635, 480408, 642723, 856398, 1136715, 1503172, 1980785

Offset: 1

N. J. A. Sloane

A partition of n is a sequence p_1, ..., p_k for some k with p_1 >= p_2 >= ... >= p_k and p_1+...+p_k=n. A partition is graphical if it is the degree sequence of a simple graph (this requires that n be even). Some authors set a(0)=1 by convention.

			a(2)=2: the graphical partitions of 4 are 2+1+1 and 1+1+1+1, corresponding to the degree sequences of the graphs V and ||.
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(1) = 1 through a(5) = 17 graphical partitions:
  (11)  (211)   (222)     (2222)      (3322)
        (1111)  (2211)    (3221)      (22222)
                (3111)    (22211)     (32221)
                (21111)   (32111)     (33211)
                (111111)  (41111)     (42211)
                          (221111)    (222211)
                          (311111)    (322111)
                          (2111111)   (331111)
                          (11111111)  (421111)
                                      (511111)
                                      (2221111)
                                      (3211111)
                                      (4111111)
                                      (22111111)
                                      (31111111)
                                      (211111111)
                                      (1111111111)
(End)

T. D. Noe, Table of n, a(n) for n = 1..860. [Terms 1 through 110 were computed by Tiffany M. Barnes and Carla D. Savage; terms 111 through 585 were computed by Axel Kohnert; terms 586 to 860 by Wang Kai, Jun 05 2016; a typo of a(547) in Number of Graphical Partitions is corrected by Wang Kai, Aug 03 2016]
Tiffany M. Barnes and Carla D. Savage, Efficient generation of graphical partitions Discrete Appl. Math. 78 (1997), no. 1-3, 17-26.
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
P. Erdős and T. Gallai, Graphs with a given degree of vertices, Mat. Lapok, 11 (1960), 264-274.
P. Erdős and L. B. Richmond, On graphical partitions Combinatorica 13 (1993), no. 1, 57-63.
Axel Kohnert, Dominance Order and Graphical Partitions, Electronic J. Combinatorics, 11 (2004).
Gerard Sierksma and Han Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory 15 (1991), no. 2, 223-231.
Eric Weisstein's World of Mathematics, Graphical partition.
Gus Wiseman, Simple graphs realizing each of the a(6) = 31 graphical partitions of 12.
Index entries for sequences related to graphical partitions

Cf. A000070, A000219, A004250, A004251, A007717, A025065, A029889, A095268, A096373, A147878, A209816, A320911, A320921, A320922.

<< MathWorld`Graphs`
Table[Count[RealizeDegreeSequence /@ Partitions[n], _Graph], {n, 2, 20, 2}]
(* second program *)
prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[prptns[#],UnsameQ@@#&]!={}&]],{n,6}] (* Gus Wiseman, Oct 26 2018 *)

A120733 Number of matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Original entry on oeis.org

1, 1, 5, 33, 281, 2961, 37277, 546193, 9132865, 171634161, 3581539973, 82171451025, 2055919433081, 55710251353953, 1625385528173693, 50800411296363617, 1693351638586070209, 59966271207156833313, 2248276994650395873861, 88969158875611127548481

Offset: 0

Vladeta Jovovic, Aug 18 2006, Aug 21 2006

nonn

The number of such matrices up to rows/columns permutations are given in A007716.
Dimensions of the graded components of the Hopf algebra MQSym (Matrix quasi-symmetric functions). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 23 2006
From Kyle Petersen, Aug 10 2016: (Start)
Number of cells in the two-sided Coxeter complex of the symmetric group. Inclusion of faces corresponds to refinement of matrices, see Section 6 of Petersen paper. The number of cells in the type B analog is given by A275787.
Also known as "two-way contingency tables" in the Diaconis-Gangolli reference. (End)

			a(2) = 5:
[1 0]   [0 1]   [1]   [1 1]   [2]
[0 1]   [1 0]   [1]
From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 33 matrices:
  [3][21][12][111]
.
  [2][20][11][11][110][101][1][10][10][100][02][011][01][01][010][001]
  [1][01][10][01][001][010][2][11][02][011][10][100][20][11][101][110]
.
  [1][10][10][10][100][100][01][01][010][01][010][001][001]
  [1][10][01][01][010][001][10][10][100][01][001][100][010]
  [1][01][10][01][001][010][10][01][001][10][100][010][100]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Sara C. Billey, M. Konvalinka, T. K. Petersen, W. Slofstra, and B. E. Tenner, Parabolic double cosets in Coxeter groups, Discrete Mathematics and Theoretical Computer Science, Submitted, 2016.
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts See IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13, page 436, Springer, Berlin.
Giulio Cerbai and Anders Claesson, Caylerian polynomials, arXiv:2310.01270 [math.CO], 2023. Mentions this sequence.
Giulio Cerbai and Anders Claesson, Enumerative aspects of Caylerian polynomials, arXiv:2411.08426 [math.CO], 2024. See pp. 3, 19.
P. Diaconis and A. Gangolli, Rectangular arrays with fixed margins, Discrete probability and algorithms (Minneapolis, MN, 1993), 15-41, IMA Vol. Math. Appl., 72, Springer, New York, 1995.
G. Duchamp, F. Hivert and J.-Y. Thibon, Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras, arXiv:math/0105065 [math.CO], 2001; Internat. J. Alg. Comp. 12 (2002), 671-717.
Loïc Foissy, Claudia Malvenuto, and Frédéric Patras, Matrix symmetric and quasi-symmetric functions and noncommutative representation theory, arXiv:2503.14417 [math.CO], 2025. See p. 11.
Masato Kobayashi, Construction of double coset system of a Coxeter group and its applications to Bruhat graphs, arXiv:1907.11801 [math.CO], 2019.
Vaclav Kotesovec, Asymptotics of the sequence A120733
E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Remark 30.
T. K. Petersen, A two-sided analogue of the Coxeter complex, arXiv:1607.00086 [math.CO], (2016).

Row sums of A261781.

Cf. A000219, A007322 (partial sums), A007716, A101370, A120732, A261838, A317533, A321515, A321585, A321586.

t1 := M -> add( add( add( (-1)^(n-j)*binomial(n, j)*((1-x)^(-j)-1)^m, j=0..n), n=0..M), m=0..M); s := series(t1(20),x,20); gfun[seriestolist](%); # N. J. A. Sloane, Jan 14 2009

a[n_] := Sum[2^(-2-r-s)*Binomial[n+r*s-1, n], {r, 0, Infinity}, {s, 0, Infinity}]; Table[Print[an = a[n]]; an, {n, 0, 19}] (* Jean-François Alcover, May 15 2012, after Vladeta Jovovic *)
Flatten[{1,Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n,k]*Sum[m!*StirlingS2[k, m],{m,k}]^2,{k,n}],{n,20}]}] (* Vaclav Kotesovec, May 07 2014 *)
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#]]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A000670(k)^2.
G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*C(n,j)*((1-x)^(-j)-1)^m.
a(n) = Sum_{r>=0,s>=0} binomial(r*s+n-1,n)/2^(r+s+2).
G.f.: Sum_{n>=0} 1/(2-(1-x)^(-n))/2^(n+1). - Vladeta Jovovic, Oct 30 2006
a(n) ~ 2^(log(2)/2-2) * n! / (log(2))^(2*n+2). - Vaclav Kotesovec, May 07 2014

More terms from N. J. A. Sloane, Jan 14 2009

A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305

Offset: 0

David S. Newman, Nov 21 2004

nonn

From Gus Wiseman, Jan 21 2019: (Start)
Also the number of semistandard Young tableaux where the rows are constant and the entries sum to n. For example, the a(8) = 19 tableaux are:
  8   44   2222   11111111
.
  1   2   11   3   111   22   1111   11   11111   1111   111111
  7   6   6    5   5     4    4      33   3       22     2
.
  1   1   11   111
  2   3   2    2
  5   4   4    3
(End)

			a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.

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

Cf. A100881, A100882, A100884.

Cf. A000085, A000219, A003293, A006951, A100471, A323582.

b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
       add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)

More terms from Vladeta Jovovic, Nov 23 2004

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

Original entry on oeis.org

1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604

Offset: 0

R. J. Mathar, Feb 11 2009

Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Ali H. Al-Saedi, Congruences for restricted plane overpartitions modulo 4 and 8, Raman. J. 48 (2) (2019) 251
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.
Mirjana Vuletic, A generalization of MacMahon's formula, Trans. Am. Math. Soc. 361 (2009) 2789-2804.

Cf. A000219, A015128, A026007, A261384, A261386, A261389.
Cf. A206622, A206623, A206624, A261519, A261520.
Cf. A285462, A285447, A285460, A285461.
Cf. A285446, A285458, A285459.
Cf. A306081.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m,2)-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, May 01 2010

Convolve A000219 with A026007.
O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021

A006330 Number of corners, or planar partitions of n with only one row and one column.

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 38, 63, 106, 170, 272, 422, 653, 986, 1482, 2191, 3218, 4666, 6726, 9592, 13602, 19122, 26733, 37102, 51232, 70292, 95989, 130356, 176246, 237120, 317724, 423840, 563266, 745562, 983384, 1292333, 1692790, 2209886, 2876132

Offset: 0

N. J. A. Sloane

nonn

The first four terms a(0), a(1), a(2), a(3) agree with sequence A000219 for unrestricted planar partitions, since the restriction does not rule anything out. For a(4) there is just one planar partition which doesn't satisfy the restriction, four 1's arranged in a square. So A000219 has fifth term 13 and here we have 12.
a(n) + A001523(n) = A000712(n). - Michael Somos, Jul 22 2003
Number of unimodal compositions of n+1 where the maximal part appears once, see example. [Joerg Arndt, Jun 11 2013]

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(4)=12 unimodal compositions of 4+1=5 where the maximal part appears once:
01:  [ 1 1 1 2 ]
02:  [ 1 1 2 1 ]
03:  [ 1 1 3 ]
04:  [ 1 2 1 1 ]
05:  [ 1 3 1 ]
06:  [ 1 4 ]
07:  [ 2 1 1 1 ]
08:  [ 2 3 ]
09:  [ 3 1 1 ]
10:  [ 3 2 ]
11:  [ 4 1 ]
12:  [ 5 ]
(End)
G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 21*x^5 + 38*x^6 + 63*x^7 + 106*x^8 + ...

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. 1, 1999; see page 77.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6).
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)
G. Kreweras, Letter to N. J. A. Sloane

Cf. A000219, A000712, A197870.
Column k=1 of A247255.
Row sums of A259100.

a[0] = 1; a[n_] := SeriesCoefficient[ Sum[x^k/Product[1 - x^i, {i, 1, k}]^2, {k, 1, n}] + 1, {x, 0, n}]; Array[a, 39, 0] (* Jean-François Alcover, Mar 13 2014 *)

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

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) - 1)\2, (-1)^k * x^((k + k^2)/2)) / eta(x + x*O(x^n))^2, n))};

G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2.
G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1 - x^k)^2. - Michael Somos, Jul 28 2003
Convolution product of A197870 and A000712. - Michael Somos, Feb 22 2015
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015

Edited and extended by Moshe Shmuel Newman, Jun 10 2003

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

cp's OEIS Frontend

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A006171 Number of factorization patterns of polynomials of degree n over integers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

PARI

Formula

A319616 Number of non-isomorphic square multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A000569 Number of graphical partitions of 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A120733 Number of matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs