A177155 -id:A177155 - cp's OEIS frontend

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

1, -1, 5, -14, 36, -97, 246, -593, 1423, -3351, 7699, -17432, 38901, -85545, 185862, -399220, 848080, -1783682, 3716584, -7675916, 15722127, -31951330, 64452707, -129102947, 256876062, -507854808, 997954125, -1949631802, 3787674152, -7319306458, 14071371173

Offset: 0

Seiichi Manyama, Apr 09 2019

sign

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

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

Product_{k>=1} (1+x^k)^((-1)^k*k^b): A083365 (b=0), A284474 (b=1), this sequence (b=2).

Cf. A027998, A177155, A266964, A294749, A294750, A307460.

nmax = 40; CoefficientList[Series[Product[(1 + x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^(4*k^2) / (1 + x^(2*k - 1))^((2*k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)

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

a(n) ~ (-1)^n * exp(2*Pi*n^(3/4)/3 + 3*Zeta(3)/(4*Pi^2)) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 09 2019

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

sign

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

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

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