A026007 -id:A026007 - cp's OEIS frontend

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

1, -1, -1, -2, 1, 0, 4, 2, 8, -2, 4, -11, -1, -25, -5, -35, 13, -26, 49, -6, 110, 6, 159, -23, 182, -141, 129, -358, 62, -640, 39, -897, 237, -1013, 771, -914, 1793, -664, 3143, -565, 4635, -1157, 5727, -3119, 6121, -7041, 5642, -13088, 5097, -20758, 5879

Offset: 0

Vaclav Kotesovec, Feb 24 2015

sign

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)
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. A000219, A026007, A026011, A073592, A262736.

Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # Vaclav Kotesovec, Mar 09 2015

nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]

{a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}
for(n=0, 100, print1(a(n), ", "))

a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 29 2015

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

Original entry on oeis.org

1, 1, 8, 35, 119, 433, 1476, 4962, 16128, 51367, 160105, 490219, 1476420, 4378430, 12805008, 36962779, 105417214, 297265597, 829429279, 2291305897, 6270497702, 17008094490, 45744921052, 122052000601, 323166712109, 849453194355, 2217289285055, 5749149331789

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5510 (terms 0..1000 from Vaclav Kotesovec)
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. 22.

Cf. A023872, A026007, A027998, A248883, A248884, A309335.

Column k=3 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^4, d=numtheory[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..35);  # Alois P. Heinz, Oct 16 2017

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

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 30 2018
Euler transform of A309335. - Georg Fischer, Nov 10 2020

A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

Original entry on oeis.org

1, 1, 4, 13, 29, 81, 188, 456, 1030, 2405, 5295, 11611, 25246, 53552, 113332, 235685, 486011, 990840, 2006567, 4018010, 7992003, 15768511, 30875424, 60060509, 116042548, 222817961, 425200270, 806991037, 1522748592, 2858792520, 5339457208, 9924370365

Offset: 0

Vaclav Kotesovec, Jan 05 2016

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n, g(n) = -n. - Seiichi Manyama, Nov 18 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - The asymptotic ratio (200000 terms)

Cf. A022629, A026007, A032302, A261562, A266964, A304210, A304211.

nmax=50; CoefficientList[Series[Product[(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
(* More efficient program: *) nmax = 50; s = 1+x; Do[s*=Sum[Binomial[k, j] * k^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 07 2016 *)

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(-d)^(1+n/d). - Seiichi Manyama, Nov 18 2017

Conjecture: log(a(n)) ~ n^(2/3) * (2*log(3*n) - 3) / (4*3^(1/3)). - Vaclav Kotesovec, May 08 2018

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)).

A078306 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2.

Original entry on oeis.org

1, 3, 10, 11, 26, 30, 50, 43, 91, 78, 122, 110, 170, 150, 260, 171, 290, 273, 362, 286, 500, 366, 530, 430, 651, 510, 820, 550, 842, 780, 962, 683, 1220, 870, 1300, 1001, 1370, 1086, 1700, 1118, 1682, 1500, 1850, 1342, 2366, 1590, 2210, 1710, 2451, 1953

Offset: 1

Vladeta Jovovic, Nov 22 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Index entries for sequences mentioned by Glaisher

Cf. A000593, A064027, A026007, A001157.

Glaisher's zeta'_i (i=0..12): A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557

a[n_] := Sum[(-1)^(n/d+1)*d^2, {d, Divisors[n]}]; Array[a, 50] (* Jean-François Alcover, Apr 17 2014 *)
Table[CoefficientList[Series[-Log[Product[1/(x^k + 1)^k, {k, 1, 90}]], {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2); \\ Michel Marcus, Jul 06 2016

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**2 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

G.f.: Sum_{n >= 1} n^2*x^n/(1+x^n).
Multiplicative with a(2^e) = (2*4^e+1)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
L.g.f.: -log(Product_{ k>0 } 1/(x^k+1)^k) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
G.f.: Sum_{n >= 1} (-1)^(n+1) * x^n*(1 + x^n)/(1 - x^n)^3. - Peter Bala, Jan 14 2021
From Vaclav Kotesovec, Aug 07 2022: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 2^(1-s)).
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / 4. (End)

A255835 G.f.: Product_{k>=1} (1+x^k)^(2*k-1).

Original entry on oeis.org

1, 1, 3, 8, 15, 34, 67, 133, 255, 486, 901, 1649, 2984, 5312, 9373, 16342, 28221, 48283, 81928, 137858, 230278, 381919, 629156, 1029933, 1675856, 2711288, 4362575, 6983196, 11122327, 17630798, 27820283, 43706461, 68375137, 106534093, 165340844, 255643289

Offset: 0

Vaclav Kotesovec, Mar 07 2015

nonn

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

Cf. A026007, A219555, A052812, A253289, A255834.

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^(1/6) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Jun 07 2018

A219555 Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.

Original entry on oeis.org

1, 2, 4, 10, 19, 38, 73, 134, 242, 430, 749, 1282, 2171, 3622, 5979, 9770, 15802, 25334, 40288, 63560, 99554, 154884, 239397, 367800, 561846, 853584, 1290107, 1940304, 2904447, 4328184, 6422164, 9489940, 13967783, 20480534, 29920277, 43557272, 63194864

Offset: 0

Alois P. Heinz, Nov 22 2012

nonn

			a(2) = 4: [(2,0)], [(1,1)], [(1,0),(0,1)], [(0,2)].

Alois P. Heinz, Table of n, a(n) for n = 0..8000 (terms n=101..1000 from Vaclav Kotesovec)

Row sums of A054242.

Cf. A026007, A052812, A005380, A255834, A255836.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
       b(n-i*j, min(n-i*j, i-1))*binomial(i+1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 19 2019

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k+1),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 07 2015 *)

a(n) = Sum_{i+j=n} [x^i*y^j] 1/2 * Product_{k,m>=0} (1+x^k*y^m).
G.f.: Product_{k>=1} (1+x^k)^(k+1). - Vaclav Kotesovec, Mar 07 2015
a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (1296*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * 3^(4/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(2 - x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Aug 11 2018

A026011 Expansion of Product_{m>=1} (1 + q^m)^(2*m).

Original entry on oeis.org

1, 2, 5, 14, 30, 68, 145, 298, 600, 1182, 2280, 4318, 8064, 14824, 26917, 48292, 85675, 150466, 261762, 451328, 771739, 1309362, 2205109, 3687904, 6127155, 10116074, 16602508, 27093582, 43974355, 71003224

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
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. 19.

Column k=2 of A277938.

Cf. A026007, A027346, A027906.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

a(n) ~ Zeta(3)^(1/6) * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/2) / (2^(2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Aug 17 2015

G.f.: exp(2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 2, 3, 0, 4, 4, 1, 10, 5, 6, 16, 6, 14, 28, 10, 32, 40, 18, 63, 60, 42, 112, 83, 84, 187, 124, 172, 300, 186, 320, 456, 302, 581, 684, 507, 982, 1004, 874, 1624, 1476, 1508, 2566, 2174, 2582, 3981, 3262, 4338, 6002, 4945, 7138, 8947, 7660

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} (1 + x^(s*k-t))^k then a(n) ~ 2^(t^2/(2*s^2) - 3/4) * s^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4 * t^2 / (1296 * s^2 * Zeta(3)) + Pi^2 * t * 2^(1/3) * 3^(2/3) * s^(2/3) * n^(1/3) / (36 * s^2 * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * s^(2/3)) ) / (3^(1/3) * s * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 12 2015

Vaclav Kotesovec, 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

Cf. A026007, A027346, A035528, A262876, A262877, A262879, A262884, A262948, A263138, A263145.

with(numtheory):
b:= n-> `if`(n<3, n-1, (p-> [0, -r, 2*r, 0, 0, 2*r+1][p]
         )(1+irem(n+3, 6, 'r'))):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

nmax=100; CoefficientList[Series[Product[(1+x^(3k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(11664*Zeta(3))) * Zeta(3)^(1/6) / (2^(25/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A027346 Expansion of Product_{m>=1} (1 + q^m)^(3*m).

Original entry on oeis.org

1, 3, 9, 28, 72, 183, 443, 1026, 2313, 5072, 10860, 22767, 46862, 94806, 188886, 371068, 719493, 1378449, 2611540, 4896291, 9090651, 16723930, 30501744, 55177932, 99048719, 176500572, 312330813, 549033172

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
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. 19.

Column k=3 of A277938.

Cf. A026007, A026011, A027906.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(3*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

a(n) ~ exp(2^(-4/3) * 3^(5/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(11/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Aug 17 2015

G.f.: exp(3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

cp's OEIS Frontend

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

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A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

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A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

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

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A078306 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2.

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A255835 G.f.: Product_{k>=1} (1+x^k)^(2*k-1).

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A219555 Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.

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A026011 Expansion of Product_{m>=1} (1 + q^m)^(2*m).

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