A321389 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k^k).

1, 2, 10, 72, 670, 7896, 113532, 1938948, 38463150, 869969602, 22098936536, 622728174288, 19271479902324, 649553475002720, 23680210649058960, 928276725059295192, 38931910620358040382, 1739307894106738293052, 82457731356894087128054, 4134332188240252347401752, 218571692793801915329820184

Offset

0,2

Comments

Convolution of A023880 and A261053.

Links

Table of n, a(n) for n=0..20.

N. J. A. Sloane, Transforms

Formula

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

a(n) ~ 2 * n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018

Maple

a:=series(mul(((1+x^k)/(1-x^k))^(k^k), k=1..100), x=0, 21): seq(coeff(a, x, n), n=0..20); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[((-1)^(k/d + 1) + 1) d^(d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

Prog

(PARI) seq(n)={Vec(exp(sum(k=1, n, sumdiv(k, d, ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

Crossrefs

Cf. A023880, A156616, A206622, A206623, A206624, A261053.

Sequence in context: A177384 A354288 A182525 * A292406 A185183 A052555

Adjacent sequences: A321386 A321387 A321388 * A321390 A321391 A321392

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 08 2018

Status

approved