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A349881 Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).

%I A349881 #20 Dec 06 2021 03:10:21
%S A349881 1,1,2,18,339,10915,663140,61264436,8044351557,1536980041573,
%T A349881 402558463751974,137204787854813174,60668198155262809815,
%U A349881 34351266752678243067591,24185207999807747975188552,20842786946335533698574605528
%N A349881 Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).
%C A349881 In general, for t>=1 and s>=0, Sum_{k=0..n} k^(t*(n-k)+s) ~ sqrt(2*Pi) * ((n + s/t)/LambertW(exp(1)*(n + s/t)))^(1/2 + (t*n + s) * (1 - 1/LambertW(exp(1)*(n + s/t)))) / sqrt(t*(1 + LambertW(exp(1)*(n + s/t)))). - _Vaclav Kotesovec_, Dec 04 2021
%H A349881 Seiichi Manyama, <a href="/A349881/b349881.txt">Table of n, a(n) for n = 0..195</a>
%F A349881 a(n) = Sum_{k=0..n} k^(4*(n-k)).
%F A349881 a(n) ~ sqrt(Pi/2) * (n/LambertW(exp(1)*n))^(1/2 + 4*n - 4*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - _Vaclav Kotesovec_, Dec 04 2021
%t A349881 a[n_] := Sum[If[k == n - k == 0, 1, k^(4*(n - k))], {k, 0, n}]; Array[a, 16, 0] (* _Amiram Eldar_, Dec 04 2021 *)
%o A349881 (PARI) a(n, s=0, t=4) = sum(k=0, n, k^(t*(n-k)+s));
%o A349881 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^4*x)))
%Y A349881 Cf. A026898, A234568, A349880.
%Y A349881 Cf. A349858.
%K A349881 nonn
%O A349881 0,3
%A A349881 _Seiichi Manyama_, Dec 03 2021