A320654 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)/(1 - x^k)).

1, 2, 8, 32, 126, 496, 1952, 7680, 30216, 118882, 467728, 1840224, 7240160, 28485616, 112073536, 440941056, 1734834302, 6825515600, 26854243752, 105655081568, 415688349456, 1635480294080, 6434618135968, 25316300481024, 99604212169632, 391881866363890, 1541816293103184

Offset: 0

Author

Ilya Gutkovskiy, Oct 18 2018

Keywords

• nonn

Comments

Invert transform of A015128.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A002448, A015128, A055887, A299108, A304969.

Programs

Maple

a:=series(1/(2-mul((1+x^k)/(1-x^k),k=1..100)),x=0,27): seq(coeff(a,x,n),n=0..26); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 26; CoefficientList[Series[1/(2 - Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[1/(2 - 1/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[PartitionsP[k - j] PartitionsQ[j], {j, 0, k}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

Formula

G.f.: 1/(2 - 1/theta_4(x)), where theta_() is the Jacobi theta function.

a(0) = 1; a(n) = Sum_{k=1..n} A015128(k)*a(n-k).