A075196 -id:A075196 - cp's OEIS frontend

A255903 Number T(n,k) of collections of nonempty multisets with a total of n objects of exactly k colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 3, 8, 5, 0, 5, 23, 33, 15, 0, 7, 56, 141, 144, 52, 0, 11, 127, 492, 848, 675, 203, 0, 15, 268, 1518, 3936, 5190, 3396, 877, 0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140, 0, 30, 1072, 11567, 57420, 154410, 240012, 216006, 104656, 21147

Offset: 0

Alois P. Heinz, Mar 10 2015

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

In the case of exactly one color (k=1) each multiset of monochrome objects is fully described by its size and a collection of sizes corresponds to an integer partition. In the case of distinct colors for all objects (k=n) every multiset collection is a set partition.

			T(3,1) = 3: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}.
T(3,2) = 8: {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}.
T(3,3) = 5: {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   8,    5;
  0,  5,  23,   33,    15;
  0,  7,  56,  141,   144,    52;
  0, 11, 127,  492,   848,   675,   203;
  0, 15, 268, 1518,  3936,  5190,  3396,   877;
  0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A255942, A255943, A255944, A255945, A255946, A255947, A255948, A255949, A255950.
Main and lower diagonals give: A000110, A255951, A255952, A255953, A255954, A255955, A255956, A255957, A255958, A255959, A255960.
Row sums give A255906.
Antidiagonal sums give A258450.
T(2n,n) gives A255907.
Cf. A075196, A317178, A326914, A326962, A327116, A327117.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i * Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12} ] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A075196(n,k-i).

Sum_{k=0..n} k * T(n,k) = A317178(n).

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

A217093 Number of partitions of n objects of 3 colors.

Original entry on oeis.org

1, 3, 12, 38, 117, 330, 906, 2367, 6027, 14873, 35892, 84657, 196018, 445746, 997962, 2201438, 4792005, 10300950, 21889368, 46012119, 95746284, 197344937, 403121547, 816501180, 1640549317, 3271188702, 6475456896, 12730032791, 24861111315, 48246729411, 93065426256

Offset: 0

Geoffrey Critzer, Sep 26 2012

a(n) is also the number of unlabeled simple graphs with n nodes of 3 colors whose components are complete graphs.
Number of (integer) partitions of n into 3 sorts of part 1, 6 sorts of part 2, 10 sorts of part 3, ..., (k+2)*(k+1)/2 sorts of part k. - Joerg Arndt, Dec 07 2014
In general the g.f. 1 / prod(n>=1, (1-x^k)^m(k) ) gives the number of (integer) partitions where there are m(k) sorts of part k. - Joerg Arndt, Mar 10 2015

			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(2) = 12 because we have: ww; wg; wb; gg; gb; bb; w + w; w + g; w + b; g + g; g + b; b + b, where the 3 colors are white w, gray g, and black b.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. Benvenuti, B. Feng, A. Hanany and Y. H. He, Counting BPS operators in gauge theories: Quivers, syzygies and plethystics, arXiv:hep-th/0608050, Aug 2006, p.42.
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

Cf. A005380, A120844, A253289, A255050, A255052, A255802, A255803, A255835, A255836, A363601, A363602.

Column k=3 of A075196.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*binomial(d+2, 2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012
with(numtheory):
series(exp(add(((1/2)*sigma[3](k) + (3/2)*sigma[2](k) + sigma[1](k))*x^k/k, k = 1..30)), x, 31):
seq(coeftayl(%, x = 0, n), n = 0..30); # Peter Bala, Jan 16 2025

nn=30; p=Product[1/(1- x^i)^Binomial[i+2,2],{i,1,nn}]; CoefficientList[Series[p,{x,0,nn}],x]

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A217093_aux(n): return sum(d*(d+1)*(d+2)>>1 for d in divisors(n,generator=True))
@lru_cache(maxsize=None)
def A217093(n): return 1 if n == 0 else (A217093_aux(n)+sum(A217093_aux(k)*A217093(n-k) for k in range(1,n)))//n # Chai Wah Wu, Mar 19 2025

G.f.: Product_{i>=1} 1/(1-x^i)^binomial(i+2,2).
EULER transform of 3, 6, 10, 15, ... .
Generally for the number of partitions of k colors the generating function is Product_{i>=1} 1/(1-x^i)^binomial(i+k-1,k-1).
a(n) ~ Pi^(1/8) * exp(1/8 + 3^4 * 5^2 * Zeta(3)^3 / (2*Pi^8) - 31*Zeta(3) / (8*Pi^2) + 5^(1/4) * Pi * n^(1/4) / 6^(3/4) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4) * Pi^5) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2) * Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4))) / (A^(3/2) * 2^(73/32) * 15^(9/32) * n^(25/32)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 08 2015
G.f.: exp(Sum_{k >= 1} ((1/2)*sigma_3(k) + (3/2)*sigma_2(k) + sigma_1(k))*x^k/k) = 1 + 3*x + 12*x^2 + 38*x^3 + 117*x^4 + .... - Peter Bala, Jan 16 2025

A255050 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+3,3).

Original entry on oeis.org

1, 4, 20, 80, 305, 1072, 3622, 11676, 36450, 110240, 324936, 935076, 2635338, 7285560, 19795370, 52930360, 139462956, 362471020, 930186694, 2358867240, 5915606398, 14680528648, 36073675792, 87816701332, 211891552280, 506981067168, 1203337174120, 2834401172172

Offset: 0

Vaclav Kotesovec, Mar 08 2015

nonn

Number of partitions of n unlabeled objects of 4 colors. - Peter Dolland, Feb 20 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio

Column k=4 of A075196.

Cf. A000292, A005380, A217093, A255052.

with(numtheory):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*binomial(d+3, 3), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50); # after Alois P. Heinz

nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+3,3],{j,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{j>=1} 1/(1-x^j)^C(j+3,3).
a(n) ~ Zeta(5)^(829/3600) * exp(11/72 - Zeta(3)/(4*Pi^2) + Zeta'(-3)/6 - 121*Zeta(3)^2 / (360*Zeta(5)) - Pi^6/(1800*Zeta(5)) + 11*Pi^8*Zeta(3) / (108000*Zeta(5)^2) - Pi^16/(194400000*Zeta(5)^3) + Pi^2 * n^(1/5)/ (6*2^(2/5) * Zeta(5)^(1/5)) - 11*Pi^4 * Zeta(3) * n^(1/5) / (900*2^(2/5)*Zeta(5)^(6/5)) + Pi^12 * n^(1/5) / (1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Zeta(3) * n^(2/5) / (6*2^(4/5) * Zeta(5)^(2/5)) - Pi^8 * n^(2/5) / (9000 * 2^(4/5) * Zeta(5)^(7/5)) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * Zeta(5)^(3/5)) + 5 * Zeta(5)^(1/5) * n^(4/5) / 2^(8/5)) / (A^(11/6) * 2^(971/1800) * 5^(1/2) * Pi * n^(2629/3600)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .
EULER transform of 1, 4, 10, 20, 35, 56, 84, ... (= A000292(n+1)). - Peter Dolland, Feb 20 2025

A255052 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+4,4).

Original entry on oeis.org

1, 5, 30, 145, 660, 2777, 11160, 42805, 158490, 568050, 1980607, 6735380, 22402610, 73022755, 233692345, 735350970, 2278153310, 6956560935, 20958613740, 62354061740, 183332498533, 533074229590, 1533842417185, 4369816273820, 12332669124455, 34495668855729

Offset: 0

Vaclav Kotesovec, Mar 08 2015

nonn

In general, if g.f. = product_{j>=1} 1/(1-x^j)^binomial(j+k-1,k-1), k>=1, then log(a(n)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio

Column k=5 of A075196.

with(numtheory):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*binomial(d+4, 4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # after Alois P. Heinz

nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+4,4],{j,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{j>=1} 1/(1-x^j)^C(j+4,4).

a(n) ~ Pi^(49/288) * exp(25/144 - 105*Zeta(3) / (8*Pi^2) + 5*Zeta'(-3)/12 + 29299*Zeta(5) / (128*Pi^4) + 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) - 72930375 * Zeta(5)^3 / (2*Pi^14) + 1063324867500 * Zeta(5)^5 / Pi^24 + 41 * 7^(1/6) * Pi * n^(1/6) / (768*3^(1/2)) - 2625 * 3^(1/2) * 7^(1/6) * Zeta(3) * Zeta(5) * n^(1/6) / (2*Pi^7) + 540225 * 3^(1/2) * 7^(1/6) * Zeta(5)^2 * n^(1/6) / (16*Pi^9) - 4740474375 * 3^(1/2) * 7^(1/6) * Zeta(5)^4 * n^(1/6) / (4*Pi^19) + 25 * 7^(1/3) * Zeta(3) * n^(1/3) / (4*Pi^2) - 735 * 7^(1/3) * Zeta(5) * n^(1/3) / (8*Pi^4) + 3969000 * 7^(1/3) * Zeta(5)^3 * n^(1/3) / Pi^14 + 7^(3/2) * Pi * n^(1/2) / (3^(3/2)*8) - 4725 * 21^(1/2) * Zeta(5)^2 * n^(1/2) / Pi^9 + 45 * 7^(2/3) * Zeta(5) * n^(2/3) / (2*Pi^4) + 2 * 3^(1/2) * Pi * n^(5/6) / (5 * 7^(1/6))) / (A^(25/12) * 2^(3/2) * 3^(625/576) * 7^(337/1728) * n^(1201/1728)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .

A075197 Number of partitions of n balls of n colors.

Original entry on oeis.org

1, 1, 6, 38, 305, 2777, 28784, 330262, 4152852, 56601345, 829656124, 12992213830, 216182349617, 3804599096781, 70540645679070, 1373192662197632, 27982783451615363, 595355578447896291, 13193917702518844859, 303931339674133588444, 7263814501407389465610

Offset: 0

Christian G. Bower, Sep 07 2002

nonn

For each integer partition of n, consider each part of size k to be a box containing k balls of up to n color. Order among parts and especially among parts of the same size does not matter. - Olivier Gérard, Aug 26 2016

			Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.
a(1) = 1:
  {{1}}
a(2) = 6 = 3+3:
  {{1,1}},{{1,2}},{{2,2}},
  {{1},{1}},{{1},{2}},{{2},{2}}
a(3) = 38 = 10+18+10:
  {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},
  {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},
  {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},
  {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},
  {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},
  {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},
  {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..400

Main diagonal of A075196.
Cf. A001700 (n balls of one color in n unlabeled boxes).
Cf. A209668 (boxes are ordered by size but not by content among a given size: order among boxes of the same size matters.),
Cf. A261783 (compositions of balls of n colors: boxes are labeled)
Cf. A252654 (lists instead of boxes : order of balls matter)
Cf. A000262 (lists instead of boxes and all n colors are used)
Cf. A255906 (the c colors used form the interval [1,c])
Cf. A255951 (the n-1 colors used form the interval [1,n-1])
Cf. A255942 (0/1 binary coloring)
Cf. A066186 (only 1 color among n = n * p(n))
Cf. A000110 (the n possible colors are used : set partitions of [n])
Cf. A005651 (the n possible colors are used and order of parts of the same size matters)
Cf. A000670 (the n possible colors are used and order of all parts matters)

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2012

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]/n]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - Ilya Gutkovskiy, May 09 2021

cp's OEIS Frontend

A255903 Number T(n,k) of collections of nonempty multisets with a total of n objects of exactly k colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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

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A217093 Number of partitions of n objects of 3 colors.

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A255050 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+3,3).

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A255052 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+4,4).

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A075197 Number of partitions of n balls of n colors.

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A270239 Number of partitions of n unlabeled objects of 6 colors.

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