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a(n) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} binomial(k, j) * (2*j + 1)^n. Row sums of A363398.
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%I #7 Jun 02 2023 08:51:40

%S 1,6,68,1280,33104,1089312,43575104,2053324800,111402371328,

%T 6839846858240,468857355838464,35494174578769920,2941165554120118272,

%U 264782344216518696960,25734702989598729256960,2685663154208346271121408,299529317622247725531725824,35554080433116190335493865472

%N a(n) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} binomial(k, j) * (2*j + 1)^n. Row sums of A363398.

%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^n * n^n / ((1 - LambertW(exp(-1))) * exp(n) * LambertW(exp(-1))^(n + 1/2)). - _Vaclav Kotesovec_, Jun 02 2023

%p a := n -> add(add(binomial(k, j)*(2*j + 1)^n, j=0..k)*2^(n-k), k=0..n):

%p seq(a(n), n = 0..17);

%t Table[Sum[2^(n-k) * Sum[Binomial[k, j] * (2*j+1)^n, {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 02 2023 *)

%Y Cf. A363398.

%K nonn

%O 0,2

%A _Peter Luschny_, Jun 02 2023