A005380 -id:A005380 - cp's OEIS frontend

A000291 Number of bipartite partitions of n white objects and 2 black ones.

2, 4, 9, 16, 29, 47, 77, 118, 181, 267, 392, 560, 797, 1111, 1541, 2106, 2863, 3846, 5142, 6808, 8973, 11733, 15275, 19753, 25443, 32582, 41569, 52770, 66757, 84078, 105555, 131995, 164566, 204450, 253292, 312799, 385285, 473183, 579722, 708353, 863553

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^2 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^2}. - Joerg Arndt, Jan 01 2024

			a(2) = 9: let p = 2 and q = 3, p^2*q^2 = 36; there are 9 factorizations: (36), (18*2), (12*3), (9*4), (9*2^2), (6*6), (6*3*2), (4*3^2), (3^2*2^2).

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
Amarnath Murthy, "Generalization of Smarandache Factor Partition introducing Smarandache Factor Partition". Smarandache Notions Journal, 1-2-3, vol. 11, 2000.
Amarnath Murthy, Program for finding out the number of Smarandache Factor Partitions. Smarandache Notions Journal, Vol. 13, 2002.
Amarnath Murthy, e-book, MS LIT format, "Ideas on Smarandache Notions".
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.9, 1.14.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
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 and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers, National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956. (Annotated scanned pages from, plus a review)

Column 2 of A054225.

Cf. A005380.

max = 40; col = 2; s1 = Series[Product[1/(1-x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}] // Normal; s2 = Series[s1, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[ a[n] , {n, 0, max}] (* Jean-François Alcover, Mar 13 2014 *)
nmax = 50; CoefficientList[Series[1/(1-x)*(1 + 1/(1-x^2))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 2 then A054225(2,n) else A054225(n,2). - Reinhard Zumkeller, Nov 30 2011
From Vaclav Kotesovec, Feb 01 2016, corrected Nov 05 2016: (Start)
a(n) = A000070(n) + A000097(n).
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 83*Pi/(24*sqrt(6*n))).
(End)

Edited by Christian G. Bower, Jan 08 2004

A060244 Triangle a(n,k) of bipartite partitions of n objects into parts >1, k of which are black.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 4, 4, 3, 2, 4, 5, 8, 8, 8, 5, 4, 4, 7, 11, 13, 13, 11, 7, 4, 7, 11, 19, 22, 26, 22, 19, 11, 7, 8, 15, 26, 35, 40, 40, 35, 26, 15, 8, 12, 22, 41, 54, 69, 70, 69, 54, 41, 22, 12, 14, 30, 56, 81, 104, 116, 116, 104, 81, 56, 30, 14, 21, 42

Offset: 0

N. J. A. Sloane, Mar 22 2001

			Series ends ... + 2*x^5 + 3*x^4*y + 4*x^3*y^2 + 4*x^2*y^3 + 3*x*y^4 + 2*y^5 + 2*x^4 + 2*x^3*y + 3*x^2*y^2 + 2*x*y^3 + 2*y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + x*y + y^2 + 1.
1;
0, 0;
1, 1, 1;
1, 1, 1, 1;
2, 2, 3, 2, 2;
...

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.

Columns 0-6: A002865, A000041, A024786, A291553, A291589, A291590, A291596.
Row sums: A060285.
Cf. A005380, A054225.

read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=2..11): SERIES2(t1,x,y,7);

max = 12; gf = Product[1/(1 - x^(i - j)*y^j), {i, 2, max}, {j, 0, i}]; se = Series[gf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[se, {x, 0, n}, {y, 0, k}]; Flatten[ Table[ t[n - k, k], {n, 0, max}, {k, 0, n}]] (* Jean-François Alcover, after Maple *)

G.f.: Product_{ i=2..infinity, j=0..i} 1/(1-x^(i-j)*y^j).

More terms from Vladeta Jovovic, Mar 23 2001

Edited by Christian G. Bower, Jan 08 2004

A120844 Number of multi-trace BPS operators for the quiver gauge theory of the orbifold C^2/Z_2.

Original entry on oeis.org

1, 3, 11, 32, 90, 231, 576, 1363, 3141, 7003, 15261, 32468, 67788, 138892, 280103, 556302, 1089991, 2108332, 4030649, 7620671, 14261450, 26431346, 48544170, 88393064, 159654022, 286149924, 509137464, 899603036, 1579014769

Offset: 0

Amihay Hanany (hanany(AT)mit.edu), Aug 25 2006

Vaclav Kotesovec, 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, 2006.
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A000203, A001157, A005380, A217093, A253289, A255271, A255802, A255803, A255834, A255835, A255836, A363601, A363602.

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-> 2*n+1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 06 2015 after Alois P. Heinz
# alternative program
with(numtheory):
series(exp(add((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

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k+1),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: exp( Sum_{n>0} (3*x^n - x^(2*n)) / (n*(1-x^n)^2) ).
a(n) ~ Zeta(3)^(7/18) * exp(1/6 - Pi^4/(864*Zeta(3)) + Pi^2 * n^(1/3)/(3 * 2^(5/3) * Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)) / (A^2 * 2^(2/9) * 3^(1/2) * Pi * n^(8/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 07 2015
From Peter Bala, Jan 16 2025: (Start)
G.f.: 1/Product_{k >= 1} (1 - x^k)^(2*k+1).
G.f.: exp(Sum_{k >= 1} (2*sigma_2(k) + sigma_1(k))*x^k/k) = 1 + 3*x + 11*x^2 + 32*x^3 + 90*x^4 + 231*x^5 + .... (End)

A255803 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+2).

Original entry on oeis.org

1, 5, 23, 86, 295, 926, 2748, 7732, 20891, 54401, 137355, 337249, 808043, 1893402, 4348634, 9805669, 21741925, 47463473, 102133056, 216841459, 454648373, 942113618, 1930779697, 3915946921, 7864385266, 15647363323, 30858285440, 60345383394, 117065924679

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(m/36 + c/6 + 1/6) * exp(m/12 - c^2 * Pi^4 / (432*m*Zeta(3)) + c * Pi^2 * n^(1/3) / (3 * 2^(4/3) * (m*Zeta(3))^(1/3)) + 3 * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(2/3)) / (A^m * 2^(c/3 + 1/3 - m/36) * 3^(1/2) * Pi^((c+1)/2) * n^(m/36 + c/6 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 08 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000219 (k), A005380 (k+1), A052847 (k-1), A120844 (2k+1), A253289 (2k-1), A255802 (2k+3), A255271 (3k+1).

Cf. A255834 - A255837, A363601, A363602.

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-> 3*n+2): seq(a(n), n=0..50); # after Alois P. Heinz
with(numtheory):
series(exp(add((3*sigma[2](k) + 2*sigma[1](k))*x^k/k, k = 1..30)), x, 31):
seq(coeftayl(%, x = 0, n), n = 0..30); # Peter Bala, Jan 16 2025

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k+2),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(7/12) * 3^(1/12) * exp(1/4 - Pi^4/(324*Zeta(3)) + Pi^2 * n^(1/3) / (3^(4/3) * (2*Zeta(3))^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A^3 * 2^(11/12) * Pi^(3/2) * n^(13/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

G.f.: exp(Sum_{k >= 1} (3*sigma_2(k) + 2*sigma_1(k))*x^k/k) = 1 + 5*x + 23*x^2 + 86*x^3 + 295*x^4 + .... - Peter Bala, Jan 16 2025

A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

Original entry on oeis.org

1, 0, 3, 8, 21, 48, 126, 288, 693, 1568, 3570, 7896, 17417, 37632, 80823, 171192, 359733, 747936, 1543192, 3155760, 6407037, 12909024, 25835649, 51359136, 101470854, 199264128, 389096028, 755591256, 1459643343, 2805471984, 5366161740, 10216161336, 19362398580

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A005380, A023871, A052847, A092348, A120844, A217093, A253289, A255271, A255802, A255803, A255835, A255836, A363602.

with(numtheory):
series(exp(add((sigma[3](k) - sigma[1](k))*x^k/k, k = 1..50)), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Jan 16 2025

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(k^2-1)))

G.f.: 1/Product_{k>=1} (1-x^k)^(k^2-1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A092348(k) * a(n-k).
G.f.: exp(Sum_{k >= 1} (sigma_3(k) - sigma_1(k))*x^k/k) = 1 + 3*x^2 + 8*x^3 + 21*x^4 + 48*x^5 + .... - Peter Bala, Jan 16 2025

Name suggested by Joerg Arndt, Jun 11 2023

A089353 Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 6, 2, 1, 5, 10, 6, 2, 1, 6, 19, 14, 6, 2, 1, 7, 28, 28, 14, 6, 2, 1, 8, 44, 52, 33, 14, 6, 2, 1, 9, 60, 93, 64, 33, 14, 6, 2, 1, 10, 85, 152, 127, 70, 33, 14, 6, 2, 1, 11, 110, 242, 228, 142, 70, 33, 14, 6, 2, 1, 12, 146, 370, 404, 272, 149, 70, 33, 14, 6, 2, 1, 13

Offset: 1

Wouter Meeussen and Vladeta Jovovic, Dec 26 2003

Also number of partitions of n objects of 2 colors into k parts, each part containing at least one black object.

			The triangle T(n,m) begins:
  n\m  1   2   3   4   5   6  7  8  9 10 11 12 ...
  1:   1
  2:   2   1
  3:   3   2   1
  4:   4   6   2   1
  5:   5  10   6   2   1
  6:   6  19  14   6   2   1
  7:   7  28  28  14   6   2  1
  8:   8  44  52  33  14   6  2  1
  9:   9  60  93  64  33  14  6  2  1
  10: 10  85 152 127  70  33 14  6  2  1
  11: 11 110 242 228 142  70 33 14  6  2  1
  12: 12 146 370 404 272 149 70 33 14  6  2  1
  ... reformatted, _Wolfdieter Lang_, Mar 09 2015

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (Ch. 11, Example 5 and Ch. 12, Example 5).
R. P. Stanley, Enumerative Combinatorics, Cambridge University Press, Vol. 2, 1999; p. 365 and Exercise 7.99, p. 484 and pp. 548-549.

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000219 (row sums), A005380, A005993 (trace 2), A050531 (trace 3), A089351 (trace 4).

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1)*x^j*
       binomial(i+j-1, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 13 2017

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[i + j - 1, j], {j, 0, n/i}]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 1, Exponent[#, x]}]& @ b[n, n];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

G.f.: Product_{k>=1} 1/(1-q*x^k)^k (with offset n=0 in x powers).

T(n+m, m) = A005380(n), n >= 1, for all m >= n. T(m, m) = 1 for m >= 1. See the Stanley reference Exercise 7.99. With offset n=0 a column for m=0 with the only non-vanishing entry T(0, 0) = 1 could be added. - Wolfdieter Lang, Mar 09 2015

Edited by Christian G. Bower, Jan 08 2004

A255271 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+1).

Original entry on oeis.org

1, 4, 17, 58, 186, 546, 1532, 4082, 10502, 26096, 63075, 148536, 342096, 771744, 1709299, 3721792, 7978972, 16860328, 35155475, 72393580, 147351112, 296657196, 591141762, 1166570452, 2281101159, 4421781894, 8500806341, 16214549920, 30696683828

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(m/36 + c/6 + 1/6) * exp(m/12 - c^2 * Pi^4 / (432*m*Zeta(3)) + c * Pi^2 * n^(1/3) / (3 * 2^(4/3) * (m*Zeta(3))^(1/3)) + 3 * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(2/3)) / (A^m * 2^(c/3 + 1/3 - m/36) * 3^(1/2) * Pi^((c+1)/2) * n^(m/36 + c/6 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 08 2015

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

Cf. A005380, A120844, A253289, A255271, A255802, A255803, A255835, A255836, A363601, A363602.

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-> 3*n+1): seq(a(n), n=0..50); # after Alois P. Heinz
with(numtheory):
series(exp(add((3*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

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

a(n) ~ Zeta(3)^(5/12) * exp(1/4 - Pi^4/(1296*Zeta(3)) + Pi^2 * n^(1/3) / (6^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A^3 * 2^(7/12) * 3^(1/12) * Pi * n^(11/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

G.f.: exp(Sum_{k >= 1} (3*sigma_2(k) + sigma_1(k))*x^k/k) = 1 + 4*x + 17*x^2 + 58*x^3 + 186*x^4 + .... - Peter Bala, Jan 16 2025

A255802 G.f.: Product_{k>=1} 1/(1-x^k)^(2*k+3).

Original entry on oeis.org

1, 5, 22, 79, 259, 777, 2201, 5911, 15239, 37865, 91224, 213741, 488759, 1093173, 2396934, 5160756, 10928181, 22787949, 46848176, 95046026, 190466354, 377295743, 739319876, 1433974869, 2754597217, 5243308562, 9894376295, 18517966608, 34386781020, 63378252332

Offset: 0

Vaclav Kotesovec, Mar 07 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A005380, A120844, A217093, A253289, A255271, A255803, A255834, A255835, A255836, A363601, A363602.

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-> 2*n+3): seq(a(n), n=0..50); # after Alois P. Heinz
with(numtheory):
series(exp(add((2*sigma[2](k) + 3*sigma[1](k))*x^k/k, k = 1..30)), x, 31):
seq(coeftayl(%, x = 0, n), n = 0..30); # Peter Bala, Jan 16 2025

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k+3),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(13/18) * exp(1/6 - Pi^4/(96*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)) / (A^2 * 2^(5/9) * 3^(1/2) * Pi^2 * n^(11/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

G.f.: exp(Sum_{k >= 1} (2*sigma_2(k) + 3*sigma_1(k))*x^k/k) = 1 + 5*x + 22*x^2 + 29*x^3 + 777*x^4 + .... - Peter Bala, Jan 16 2025

A000412 Number of bipartite partitions of n white objects and 3 black ones.

Original entry on oeis.org

3, 7, 16, 31, 57, 97, 162, 257, 401, 608, 907, 1325, 1914, 2719, 3824, 5313, 7316, 9973, 13495, 18105, 24132, 31938, 42021, 54948, 71484, 92492, 119120, 152686, 194887, 247693, 313613, 395547, 497154, 622688, 777424, 967525, 1200572, 1485393, 1832779, 2255317

Offset: 0

N. J. A. Sloane

nonn

Number of ways to factor p^n*q^3 where p and q are distinct primes.
Number of Gaussian partitions of n+3*i or 3+n*i where a "Gaussian partition" is a way of writing a Gaussian integer with nonnegative parts as a sum of Gaussian integers with nonnegative parts, imaginary numbers and real numbers. For k = 3+1*i (where i is the imaginary unit), the a(1)=7 ways to write k (where parentheses represent a complex number and a lack of them represents a sum of a real and imaginary number) would be 3+i, (3+i), 2+1+i, (2+i)+1, (1+i)+2, 1+1+1+i, (1+i)+1+1. - Yali Harrary, Nov 20 2022
a(n) is the number of multiset partitions of the multiset {r^n, s^3}. - Joerg Arndt, Jan 01 2024

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..100 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953), pp. 72-83.
F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (annotated scanned pages from, plus a review).

Column 3 of A054225.

Cf. A005380.

max = 40; col = 3; s1 = Series[Product[1/(1-x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}] // Normal; s2 = Series[s1, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[ a[n] , {n, 0, max}] (* Jean-François Alcover, Mar 13 2014 *)
nmax = 50; CoefficientList[Series[(3 + x - x^2 - 2*x^3 - x^4 + x^5)/((1-x)*(1-x^2)*(1-x^3)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

a(n) = if n <= 3 then A054225(3,n), otherwise a(n) = A054225(n,3). - Reinhard Zumkeller, Nov 30 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Feb 01 2016
a(n) = A000098(n) + A000070(n) + A014153(n). - Yali Harrary, Nov 20 2022

Edited by Christian G. Bower, Jan 08 2004

A381891 Triangle read by rows: T(n,k) is the number of partitions of a 2-colored set of n objects into at most k parts with 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 3, 6, 0, 4, 10, 14, 0, 5, 19, 28, 33, 0, 6, 28, 52, 64, 70, 0, 7, 44, 93, 127, 142, 149, 0, 8, 60, 152, 228, 272, 290, 298, 0, 9, 85, 242, 404, 507, 561, 582, 591, 0, 10, 110, 370, 672, 904, 1034, 1098, 1122, 1132, 0, 11, 146, 546, 1100, 1568, 1870, 2027, 2101, 2128, 2139

Offset: 0

Peter Dolland, Mar 09 2025

The 1-color case is Euler's table A026820.

			Triangle begins:
  1;
  0, 2;
  0, 3,  6;
  0, 4, 10,  14;
  0, 5, 19,  28,  33;
  0, 6, 28,  52,  64,  70;
  0, 7, 44,  93, 127, 142, 149;
  0, 8, 60, 152, 228, 272, 290, 298;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A005380, A026820.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (n+1)*x^n,
      add(b(n-i*j, min(n-i*j, i-1))*binomial(i+j, j)*x^j, j=0..n/i)))
    end:
T:= proc(n, k) option remember;
      `if`(k<0, 0, T(n, k-1)+coeff(b(n$2), x, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 09 2025

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
def a381891_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( k + p[k], p[k])
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

T(1,k) = k + 1.

T(n,n) = A005380(n).

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