A266857 -id:A266857 - cp's OEIS frontend

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.

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

A261562 Expansion of Product_{k>=1} (1 + 2*x^k)^k.

Original entry on oeis.org

1, 2, 4, 14, 24, 58, 124, 238, 480, 922, 1764, 3238, 6008, 10794, 19292, 34166, 59504, 103042, 176452, 299958, 505240, 845570, 1403324, 2315118, 3794640, 6180370, 10009540, 16121374, 25829512, 41171690, 65320956, 103140062, 162149488, 253823178, 395698276

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 2001 terms from Vaclav Kotesovec)

Cf. A026007, A032302, A261561, A261563, A266857, A266891.

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 21 2018

nmax = 50; CoefficientList[Series[Product[(1 + 2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 50; s = 1+2*x; Do[s*=Sum[Binomial[k, j]*2^j*x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 08 2016 *)

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-2)^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-2)^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

a(n) ~ c^(1/6) * exp(3^(2/3)*c^(1/3)*n^(2/3)/2) / (3^(3/4)*sqrt(2*Pi)*n^(2/3)), where c = Pi^2*log(2) + log(2)^3 - 6*polylog(3, -1/2) = 10.00970018379942727227807189532511265744588249928680712064... . - Vaclav Kotesovec, Jan 04 2016

A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

Original entry on oeis.org

1, 1, 4, 27, 80, 400, 1908, 6223, 31296, 116478, 450100, 1828915, 7360848, 26906828, 95776772, 403908975, 1421758720, 5072014447, 18481180644, 68350964211, 246180936400, 827642046294, 2958748580084, 10294629775620, 36607347335232, 120800714172500, 407951731319860, 1405943613730899

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

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

Cf. A026007, A261562, A266857, A266891, A291698, A297322, A298985, A298986, A298988.

Table[SeriesCoefficient[Product[(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

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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.

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A261562 Expansion of Product_{k>=1} (1 + 2*x^k)^k.

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A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

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