A120844 -id:A120844 - cp's OEIS frontend

A217093 Number of partitions of n objects of 3 colors.

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

A253289 G.f.: Product_{k>=1} 1/(1-x^k)^(2*k-1).

Original entry on oeis.org

1, 1, 4, 9, 22, 46, 103, 208, 431, 849, 1671, 3195, 6079, 11321, 20937, 38146, 68931, 123121, 218212, 383019, 667425, 1153544, 1980268, 3375394, 5717773, 9624541, 16108496, 26807662, 44379189, 73089219, 119789926, 195401275, 317309532, 513025167, 826000651

Offset: 0

Vaclav Kotesovec, Mar 07 2015

a(n) is the number of partitions of n where there are 2*k-1 sorts of parts k. - Joerg Arndt, Aug 15 2020

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

Cf. A005380, A120844, A217093, 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-> 2*n-1): seq(a(n), n=0..50); # after Alois P. Heinz
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]
(* Using EulerTransforms from 'Transforms'. *)
Prepend[EulerTransform[Table[2k + 1, {k, 0, 20}]], 1] (* Peter Luschny, Aug 15 2020 *)

a(n) ~ 2^(1/9) * Zeta(3)^(1/18) * 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 * 3^(1/2) * n^(5/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Jun 07 2018
Euler transform of A005408 (the odd numbers). - Georg Fischer, Aug 15 2020
G.f.: exp(Sum_{k >= 1} (2*sigma_2(k) - sigma_1(k))*x^k/k) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + .... - Peter Bala, Jan 16 2025

A255834 G.f.: Product_{k>=1} (1+x^k)^(2*k+1).

Original entry on oeis.org

1, 3, 8, 23, 55, 129, 291, 627, 1317, 2697, 5398, 10589, 20421, 38743, 72452, 133724, 243792, 439496, 784070, 1385195, 2424971, 4209094, 7247141, 12383496, 21008559, 35398548, 59259781, 98595110, 163077878, 268221706, 438791204, 714142139, 1156552537

Offset: 0

Vaclav Kotesovec, Mar 07 2015

nonn

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

Cf. A026007, A120844, A219555, A255835, A255836.

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

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (2592*Zeta(3)) + Pi^2 * n^(1/3) / (12*(3*Zeta(3))^(1/3)) + 3^(4/3)/2*Zeta(3)^(1/3) * n^(2/3)) / (2^(7/6)* 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

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

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

cp's OEIS Frontend

A217093 Number of partitions of n objects of 3 colors.

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A253289 G.f.: Product_{k>=1} 1/(1-x^k)^(2*k-1).

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A255834 G.f.: Product_{k>=1} (1+x^k)^(2*k+1).

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

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

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

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

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