A255528 -id:A255528 - cp's OEIS frontend

A278710 Convolution square of A255528.

1, -2, -1, -2, 7, 2, 10, -8, 5, -40, -4, -54, 52, -30, 162, -12, 292, -142, 270, -576, 168, -1228, 305, -1702, 1435, -1664, 3839, -1444, 7303, -2752, 10117, -8420, 11065, -20714, 11066, -38702, 17057, -57276, 40310, -69898, 94138, -77014, 181926, -97480

Offset: 0

Seiichi Manyama, Apr 11 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>0} 1/(1 + x^k)^(k*m): A026011 (m=-2), A255528 (m=1), this sequence (m=2), A279031 (m=3), A279411 (m=4).

G.f.: Product_{k>0} 1/(1 + x^k)^(k*2).
a(n) ~ (-1)^n * exp(-1/6 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^2 * Zeta(3)^(1/9) / (2^(11/18) * sqrt(3*Pi) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017
G.f.: exp(2*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

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

A073592 Euler transform of negative integers.

Original entry on oeis.org

1, -1, -2, -1, 0, 4, 4, 7, 3, -2, -9, -17, -25, -24, -13, -1, 32, 61, 97, 111, 112, 74, 8, -108, -243, -392, -512, -569, -542, -358, -33, 473, 1078, 1788, 2395, 2865, 2955, 2569, 1496, -245, -2751, -5783, -9121, -12299, -14739, -15806, -14719, -10930, -3813, 6593, 20284, 36139, 53081, 68620, 80539

Offset: 0

Vladeta Jovovic, Aug 28 2002

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1/A(x) is g.f. for A000219.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
E. M. Wright, Coefficients of a reciprocal generating function, Quart. J. Math. 17 (1) (1966) 39-43, ADS Abstracts.
N. J. A. Sloane, Transforms

Column k=1 of A283272.

Cf. A000219, A001157, A001478, A026007, A156616, A255528.

a:= proc(n) option remember; `if`(n=0, 1, -add(
      numtheory[sigma][2](j)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 12 2015

nmax=50; CoefficientList[Series[Exp[Sum[-x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 02 2015 *)
a[n_]:= a[n] = -1/n*Sum[DivisorSigma[2,k]*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 02 2015 *)

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n)
print([b(n) for n in range(55)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k>0} (1-x^k)^k.
a(n) = -1/n*Sum_{k=1..n} sigma[2](k)*a(n-k).
G.f.: exp( Sum_{n>=1} -sigma_2(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017

A262736 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1).

Original entry on oeis.org

1, 1, 0, 3, 3, 5, 8, 10, 22, 25, 41, 57, 88, 126, 168, 261, 351, 512, 685, 984, 1357, 1865, 2566, 3485, 4838, 6459, 8832, 11831, 16056, 21404, 28660, 38259, 50875, 67613, 89161, 118184, 155321, 204609, 267708, 351125, 458331, 597740, 777590, 1010020, 1310390

Offset: 0

Vaclav Kotesovec, Sep 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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. 20.

Cf. A026007, A026011, A255528, A262811, A292038.

Cf. A262924, A285292, A285293.

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3^(4/3) * (Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * Zeta(3)^(1/6) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)).

A284896 Expansion of Product_{k>=1} 1/(1+x^k)^(k^2) in powers of x.

Original entry on oeis.org

1, -1, -3, -6, 0, 11, 42, 63, 73, -45, -267, -720, -1095, -1239, -66, 2794, 8757, 16017, 22885, 19634, -2359, -61979, -161867, -302190, -421971, -432051, -126712, 690578, 2278273, 4584989, 7269985, 8965464, 7515373, -845659, -19930400, -53474765, -100195759

Offset: 0

Seiichi Manyama, Apr 05 2017

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This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n^2, g(n) = -1. - Seiichi Manyama, Nov 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A027998, A281590, A281591.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), this sequence (m=2), A284897 (m=3), A284898 (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^2) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^2))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078307(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

G.f.: exp(Sum_{k>=1} (-1)^k*x^k*(1 + x^k)/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 30 2018

A284628 Expansion of Product_{k>=1} 1/(1+x^(2k-1))^(2k-1).

Original entry on oeis.org

1, -1, 1, -4, 4, -9, 15, -22, 37, -56, 92, -133, 210, -310, 466, -696, 1013, -1495, 2160, -3141, 4495, -6462, 9172, -13024, 18387, -25840, 36213, -50500, 70280, -97302, 134522, -185105, 254245, -347938, 475036, -646676, 878145, -1189468, 1607095, -2166672, 2913794

Offset: 0

Seiichi Manyama, Apr 15 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A255528, A262736, A262811.

nmax = 50; CoefficientList[Series[Product[1/(1+x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

a(n) = (-1)^n * A262811(n).

A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 4, 3, 0, 0, 6, 6, 0, 4, 7, 6, 3, 8, 8, 6, 6, 4, 21, 20, 4, 1, 34, 34, 2, 8, 23, 44, 28, 19, 18, 54, 54, 18, 56, 65, 46, 25, 100, 94, 38, 42, 85, 169, 107, 56, 69, 226, 194, 62, 111, 194, 241, 125, 215, 246, 258, 207, 283, 437, 292

Offset: 0

Vaclav Kotesovec, Apr 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. A000219, A026007, A027998, A033461, A255528, A266891, A284896, A285047.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s*=Sum[Binomial[k, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

a(n) ~ exp(sqrt(n/6)*Pi) / (2^(11/6) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 15 2017

A279031 Expansion of Product_{k>0} 1/(1 + x^k)^(k*3).

Original entry on oeis.org

1, -3, 0, -1, 15, -3, 8, -42, 6, -83, 81, -39, 316, -90, 420, -603, 363, -1656, 625, -2556, 2877, -2599, 7818, -3483, 13886, -11049, 17040, -31493, 20196, -63876, 39244, -96453, 105891, -120431, 243333, -164100, 440873, -327387, 643968, -765115, 840207

Offset: 0

Seiichi Manyama, Apr 11 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>0} 1/(1 + x^k)^(k*m): A027346 (m=-3), A255528 (m=1), A278710 (m=2), this sequence (m=3), A279411 (m=4).

a(n) ~ (-1)^n * exp(-1/4 + 2^(-5/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^3 * Zeta(3)^(1/12) / (2^(2/3) * 3^(5/12) * sqrt(Pi) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(3*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A279411 Expansion of Product_{k>0} 1/(1 + x^k)^(k*4).

Original entry on oeis.org

1, -4, 2, 0, 23, -20, 2, -88, 63, -96, 318, -104, 626, -844, 504, -2472, 1525, -3704, 6184, -4288, 15284, -10736, 23254, -35792, 30228, -84544, 60974, -139240, 176658, -190108, 418940, -320976, 755332, -773524, 1111678, -1847304, 1669046, -3634296

Offset: 0

Seiichi Manyama, Apr 11 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=4 of A279928.

Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), this sequence (m=4), A279932 (m=5).

a(n) ~ (-1)^n * exp(-1/3 + 3/2 * Zeta(3)^(1/3) * n^(2/3)) * A^4 * Zeta(3)^(1/18) / (sqrt(6*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(4*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 1, 0, 1, -5, 2, -1, 7, 0, 0, 1, -6, 5, 0, 15, 2, 4, 0, 1, -7, 9, 0, 23, -3, 10, 2, 0, 1, -8, 14, -2, 30, -20, 8, -8, 8, 0, 1, -9, 20, -7, 36, -51, 2, -42, 5, -2, 0, 1, -10, 27, -16, 42, -96, 5, -88, 6

Offset: 0

Seiichi Manyama, Apr 11 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -1,  0,  2, ...
   0, -2, -2, -1,  0, ...
   0,  1,  7, 15, 23, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000007, A255528, A278710, A279031, A279411, A279932.
Main diagonal gives A281266.
Antidiagonal sums give A299212.
Cf. A255961, A277938.

G.f. of column k: Product_{j>=1} 1/(1+x^j)^(j*k).

cp's OEIS Frontend

A278710 Convolution square of A255528.

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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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A073592 Euler transform of negative integers.

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A262736 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(2*k-1).

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A284896 Expansion of Product_{k>=1} 1/(1+x^k)^(k^2) in powers of x.

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A284628 Expansion of Product_{k>=1} 1/(1+x^(2*k-1))^(2*k-1).

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A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

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A279031 Expansion of Product_{k>0} 1/(1 + x^k)^(k*3).

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A279411 Expansion of Product_{k>0} 1/(1 + x^k)^(k*4).

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A262736 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1).

A284628 Expansion of Product_{k>=1} 1/(1+x^(2k-1))^(2k-1).