A307460 -id:A307460 - 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

A307514 Expansion of Product_{k>=1} (1-x^k)^((-1)^k*k^k).

Original entry on oeis.org

1, 1, -3, 24, -226, 2791, -42467, 761826, -15714798, 366401751, -9528266885, 273439284005, -8584541521286, 292695692569785, -10771202678289501, 425538242701632216, -17964593967281888258, 807094224863059707077, -38449142619220645357810, 1935991142823285710574298

Offset: 0

Seiichi Manyama, Apr 12 2019

sign

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

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A010054, A266964, A281781, A283499, A307460, A307497, A307504.

nmax=20; CoefficientList[Series[Product[(1-x^k)^((-1)^k*k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 12 2019 *)

N=20; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^((-1)^k*k^k)))

a(n) ~ -(-1)^n * n^n * (1 - exp(-1)/n - (exp(-1)/2 + 3*exp(-2))/n^2). - Vaclav Kotesovec, Apr 12 2019

cp's OEIS Frontend

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

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