A265949 Expansion of Product_{k>=1} (1 + k^k*x^k).

1, 1, 4, 31, 283, 3489, 50913, 890635, 17891170, 409850236, 10494427982, 297780829216, 9261266862273, 313453533534739, 11464487066049791, 450644378868285130, 18942868694407904729, 847930346323808122469, 40266107916200371331007, 2021842180288047801103956

Offset: 0

Author

Vaclav Kotesovec, Dec 19 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A023882, A292190, A292305, A292306.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

Maple

seq(coeff(series(mul((1+k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

Mathematica

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

PARI

m=30; x='x+O('x^m); Vec(prod(k=1, m, (1+k^k*x^k))) \\ G. C. Greubel, Oct 31 2018

Formula

a(n) ~ n^n * (1 + exp(-1)/n + ((1/2)*exp(-1) + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d + 1)*d^(k+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018