A262736 -id:A262736 - 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

A262811 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, 3910741

Offset: 0

Vaclav Kotesovec, Oct 03 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

Cf. A000219, A003293, A035528, A161870, A262736, A292038.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d::even, 0, d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

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

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

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 09 2017

A285069 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

Offset: 0

Seiichi Manyama, Apr 09 2017

sign

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

Cf. A050999, A262736, A262811.

CoefficientList[Series[Product[(1 - x^(2k-1))^(2k-1), {k, 50}], {x, 0, 50}], x] (* Indranil Ghosh, Apr 09 2017 *)

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k/k))) \\ Seiichi Manyama, Oct 31 2017

a(n) = (-1)^n * A262736(n).
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0.
a(n) ~ (-1)^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)). - Vaclav Kotesovec, Nov 09 2017

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 2, 2, 5, 10, 13, 25, 35, 57, 87, 134, 211, 306, 458, 684, 996, 1465, 2129, 3073, 4411, 6288, 8977, 12707, 17913, 25185, 35231, 49078, 68228, 94490, 130408, 179425, 246121, 336681, 459239, 624842, 847986, 1147728, 1549773, 2087972, 2806455, 3764136

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262948 and A262949.

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

Cf. A262884, A262878, A262879, A262923, A261612, A262928, A262948, A262949.

Cf. A262736, A285292, A285293.

nmax=60; CoefficientList[Series[Product[(1 + x^(3*k-1))^(3*k-1)*(1 + x^(3*k-2))^(3*k-2), {k, 1, nmax}], {x, 0, nmax}], x]

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

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

sign

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).

A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 6, 15, 9, 0, 4, 40, 36, 0, 17, 71, 90, 36, 64, 100, 180, 144, 96, 274, 394, 300, 148, 740, 820, 480, 472, 1150, 1851, 1341, 1146, 1318, 3880, 3540, 1704, 3017, 6455, 7134, 3780, 7822, 9574, 12180, 10304, 12057, 19750, 22485, 20558, 15910, 43076, 43236, 31104, 33742, 66895

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Number of partitions of n into distinct squares, where k^2 different parts of size k^2 are available (1a, 4a, 4b, 4c, 4d, ...).

			a(8) = 6 because we have [4a, 4b], [4a, 4c], [4a, 4d], [4b, 4c], [4b, 4d] and [4c, 4d].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000290, A026007, A027998, A033461, A262736, A281790, A284896, A291655, A291692.

nmax = 100; CoefficientList[Series[Product[(1 + x^k^2)^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^2, 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] (* Vaclav Kotesovec, Aug 28 2017 *)

G.f.: Product_{k>=1} (1 + x^A000290(k))^A000290(k).

a(n) ~ exp(5 * 2^(-9/5) * 3^(-3/5) * (9-4*sqrt(2))^(1/5) * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5)) * 3^(1/5) * (2*sqrt(2)-1)^(1/5) * Zeta(5/2)^(1/5) / (2^(9/10) * sqrt(5) * Pi^(2/5) * n^(7/10)). - Vaclav Kotesovec, Aug 29 2017

A262948 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1).

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 10, 8, 5, 26, 11, 28, 62, 24, 101, 111, 77, 260, 202, 268, 583, 382, 761, 1165, 847, 1940, 2198, 2061, 4346, 4084, 5078, 9039, 7844, 11978, 17620, 15721, 26648, 33219, 32894, 56000, 61494, 69653, 111884, 114265, 146557, 214864, 214967

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

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

Cf. A262736, A262878, A262884, A262924, A262928, A262949.

nmax=60; CoefficientList[Series[Product[(1 + x^(3*k-1))^(3*k-1),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(7/12) * sqrt(3*Pi) * n^(2/3)).

A285292 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(4k))^(4k).

Original entry on oeis.org

1, 1, 2, 5, 4, 12, 20, 29, 53, 80, 127, 199, 311, 468, 715, 1079, 1621, 2402, 3541, 5210, 7574, 11046, 15926, 22917, 32804, 46766, 66419, 93936, 132331, 185830, 260144, 362752, 504573, 699376, 966842, 1332721, 1832217, 2512209, 3435932, 4687884, 6380911

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)

Cf. A262736, A262924, A285293.

nmax = 50; CoefficientList[Series[Product[(1+x^k)^k/(1+x^(4*k))^(4*k), {k, 1, nmax}], {x, 0, nmax}], x]

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

A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).

Original entry on oeis.org

1, 1, -2, 1, 2, -2, 0, -5, 10, 1, -15, 10, -1, 18, -39, 4, 50, -24, -14, -69, 165, -70, -83, -20, 154, 161, -550, 313, 55, 410, -960, 102, 1074, -406, -506, -1344, 3581, -1791, -833, -1833, 4995, 205, -6993, 2982, 2461, 7649, -19791, 9495, 4986, 9581, -26745, 0

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A224364, A262736, A278710, A281683, A284474.

N:= 100: # to get a(0)..a(N)
P:= mul((1+x^(2*k-1))^(2*k-1)/(1+x^(2*k))^(2*k),k=1..N/2):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Apr 16 2017

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, Jun 20 2018

A285288 Expansion of Product_{k>=0} (1 + x^(4k+1))^(4k+1).

Original entry on oeis.org

1, 1, 0, 0, 0, 5, 5, 0, 0, 9, 19, 10, 0, 13, 58, 55, 10, 17, 118, 191, 95, 26, 223, 512, 400, 116, 362, 1175, 1329, 564, 609, 2368, 3593, 2218, 1246, 4402, 8600, 7118, 3433, 7792, 18503, 19778, 10702, 13924, 37009, 49017, 32097, 27141, 69629, 111251, 88972

Offset: 0

Seiichi Manyama, Apr 16 2017

nonn

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

Product_{k>=0} (1 + x^(m*k+1))^(m*k+1): A262736 (m=2), A262949 (m=3), this sequence (m=4).

Cf. A285070, A285287.

nmax = 50; CoefficientList[Series[Product[(1 + x^(4*k-3))^(4*k-3), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

a(n) = (-1)^n * A285070(n).

a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 16 2017

cp's OEIS Frontend

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

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A262811 Expansion of Product_{k>=1} 1/(1-x^(2*k-1))^(2*k-1).

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A285069 Expansion of Product_{k>=1} (1 - x^(2*k-1))^(2*k-1).

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A262924 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1) * (1 + x^(3*k-2))^(3*k-2).

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

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A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

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A262948 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1).

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A285292 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(4*k))^(4*k).

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A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).

A285069 Expansion of Product_{k>=1} (1 - x^(2k-1))^(2k-1).

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

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

A262948 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1).

A285292 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(4k))^(4k).

A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).

A285288 Expansion of Product_{k>=0} (1 + x^(4k+1))^(4k+1).