A321389 - OEIS

%I #11 Apr 02 2019 05:51:24

%S 1,2,10,72,670,7896,113532,1938948,38463150,869969602,22098936536,

%T 622728174288,19271479902324,649553475002720,23680210649058960,

%U 928276725059295192,38931910620358040382,1739307894106738293052,82457731356894087128054,4134332188240252347401752,218571692793801915329820184

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

%C Convolution of A023880 and A261053.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

%F a(n) ~ 2 * n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - _Vaclav Kotesovec_, Nov 09 2018

%p 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

%t nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t 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}]

%o (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

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 08 2018