A284467 -id:A284467 - cp's OEIS frontend

A284474 Expansion of Product_{k>=1} (1 + x^(2k))^(2k)/(1 + x^(2k-1))^(2k-1).

1, -1, 3, -6, 11, -22, 42, -74, 131, -231, 395, -669, 1122, -1851, 3029, -4915, 7891, -12572, 19881, -31203, 48657, -75391, 116096, -177792, 270822, -410394, 618905, -929052, 1388403, -2066140, 3062270, -4520912, 6649463, -9745072, 14232278, -20716355, 30057438

Offset: 0

Seiichi Manyama, Apr 15 2017

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Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A026011, A281781, A284467, A284628.

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

a(n) ~ (-1)^n * exp(-1/12 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * A * (7*Zeta(3))^(5/36) / (2^(10/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 17 2017

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

A295834 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^(k+1)/k).

Original entry on oeis.org

1, 1, -1, -1, 11, 19, -311, -1919, 20201, 154169, -1363249, -14236289, 140759299, 1213688059, -33239720359, -257577468511, 11707385639249, 119005356808561, -3416942071608929, -43117983466829441, 893917358612502011, 13133282766425234531, -411010168576899605911, -7970128344774479644991

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

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			E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...)/((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...) = 1 + x - x^2/2! - x^3/3! + 11*x^4/4! + 19*x^5/5! - 311*x^6/6! - 1919*x^7/7! + ...

Cf. A028342, A168243, A206303, A284467, A294356, A295792, A295833.

a:=series(mul((1+x^k)^((-1)^(k+1)/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((-1)^(k+1)/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A307484 Expansion of Product_{k>=1} 1/(1+x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, 1, -4, 5, 3, -17, 33, -61, 67, 63, -392, 803, -1070, 898, 482, -4449, 11362, -18630, 21105, -11067, -24871, 103562, -227004, 359040, -417697, 266106, 312987, -1578543, 3635615, -6157911, 8155892, -7689028, 1502546, 14707881, -44539735, 87849728, -136927058, 171008704

Offset: 0

Seiichi Manyama, Apr 10 2019

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

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

Product_{k>=1} 1/(1+x^k)^((-1)^k*k^b): A029838 (b=0), A284467 (b=1), this sequence (b=2).

Cf. A177155, A307462.

m = 37; CoefficientList[Series[Product[1/(1+x^k)^((-1)^k*k^2), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 + x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 14 2021 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^((-1)^k*k^2)))

cp's OEIS Frontend

A284474 Expansion of Product_{k>=1} (1 + x^(2k))^(2k)/(1 + x^(2k-1))^(2k-1).

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A295834 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^(k+1)/k).

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A307484 Expansion of Product_{k>=1} 1/(1+x^k)^((-1)^k*k^2).

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cp's OEIS Frontend

A284474 Expansion of Product_{k>=1} (1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1).

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A295834 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^(k+1)/k).

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A307484 Expansion of Product_{k>=1} 1/(1+x^k)^((-1)^k*k^2).

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A284474 Expansion of Product_{k>=1} (1 + x^(2k))^(2k)/(1 + x^(2k-1))^(2k-1).