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a(n) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} binomial(k, j) * (j + 1)^n. Row sums of A363399.
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%I #7 Jun 02 2023 08:51:49

%S 1,5,32,302,3904,64272,1286144,30313712,822571008,25258008320,

%T 865863532544,32779942009344,1358320701014016,61149815860711424,

%U 2971951570679234560,155090406558662064128,8649258967534890123264,513370937392454603833344

%N a(n) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} binomial(k, j) * (j + 1)^n. Row sums of A363399.

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

%p a := n -> add(add(binomial(k, j)*(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] * (j+1)^n, {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 02 2023 *)

%Y Cf. A363399.

%K nonn

%O 0,2

%A _Peter Luschny_, Jun 02 2023