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

0,3

Comments

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

Links

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

Formula

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

Mathematica

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 *)

Prog

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

Crossrefs

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.

Sequence in context: A052951 A048745 A372304 * A292170 A224716 A127980

Adjacent sequences: A307459 A307460 A307461 * A307463 A307464 A307465

Keyword

sign

Author

Seiichi Manyama, Apr 09 2019

Status

approved