A320564 Expansion of Product_{k>=1} (1 + x^k/(1 - x)^k)^k.

1, 1, 3, 10, 30, 87, 249, 705, 1974, 5471, 15032, 40997, 111079, 299151, 801139, 2134251, 5657895, 14930596, 39232009, 102673794, 267692321, 695440442, 1800582809, 4646964755, 11956293758, 30673060344, 78470890246, 200218512582, 509557661691, 1293664233400, 3276659862518

Offset: 0

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

• nonn

Comments

First differences of the binomial transform of A026007.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A026007, A129519, A294502, A320563.

Programs

Maple

seq(coeff(series(mul((1+x^k/(1-x)^k)^k,k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 15 2018

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 + x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (1 - x)^k x^k/(k ((1 - x)^k - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*(1 - x)^k*x^k/(k*((1 - x)^k - x^k)^2)).

a(n) ~ Zeta(3)^(1/6) * 2^(n - 13/12) * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4 + (3*Zeta(3))^(2/3) * n^(1/3)/8 - Zeta(3)/16) / (3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 15 2018