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a(n) = Sum_{k=0..n} k^(k*(n-k)).
8

%I #20 Dec 06 2021 03:09:34

%S 1,2,3,7,46,1052,88603,27121965,37004504306,198705527223758,

%T 5595513387083114571,686714367475480207331583,

%U 468422339816915120237104999422,1664212116512828935888786624225704856,31295654819650678010096952493864470025103251

%N a(n) = Sum_{k=0..n} k^(k*(n-k)).

%H Seiichi Manyama, <a href="/A349893/b349893.txt">Table of n, a(n) for n = 0..52</a>

%F G.f.: Sum_{k>=0} x^k/(1 - k^k * x).

%F log(a(n)) ~ n^2*log(n)/4 * (1 - log(2)/log(n) + 1/(4*log(n)^2)). - _Vaclav Kotesovec_, Dec 05 2021

%t Table[1 + Sum[k^(k*(n - k)), {k, 1, n}], {n, 0, 16}] (* _Vaclav Kotesovec_, Dec 05 2021 *)

%o (PARI) a(n) = sum(k=0, n, k^(k*(n-k)));

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^k*x)))

%Y Cf. A117402, A349880, A349881, A349886, A349894.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 04 2021