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%I #5 Jan 23 2021 10:47:10
%S 1,4,139,11928,1909787,491329088,185373016419,96425597012608,
%T 66139668570414571,57840395870803141632,62813828698519808489915,
%U 82933938539372018962724864,130828514220436815006398809563,243020960809424084526916839817216,525038425527430196237626528753654867
%N a(n) = (1/2) * Sum_{k>=0} (k*(k + n))^n / 2^k.
%F a(n) = Sum_{k=0..n} binomial(n,k) * A000670(2*n-k) * n^k.
%t Table[(1/2) Sum[(k (k + n))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
%t Join[{1}, Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0] n^k, {k, 0, n}], {n, 1, 14}]]
%Y Cf. A000670, A098696, A249938, A292916, A340822.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 23 2021
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