A360474 E.g.f. satisfies A(x) = exp( x * A(x)^2 * exp(x * A(x)^2) ).
1, 1, 7, 94, 1921, 53036, 1849789, 78070462, 3869909537, 220427550712, 14188370562901, 1018570771664546, 80692202644742737, 6992855583524143204, 658076908751441373965, 66833181471569822199886, 7285736943975575120653249, 848589321771735983890457072
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Keywords
Programs
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Mathematica
Join[{1}, Table[Sum[k^(n-k) * (2*n+1)^(k-1) * Binomial[n,k], {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Feb 17 2023 *) -
PARI
a(n) = sum(k=0, n, k^(n-k)*(2*n+1)^(k-1)*binomial(n, k));
Formula
a(n) = Sum_{k=0..n} k^(n-k) * (2*n+1)^(k-1) * binomial(n,k).
a(n) ~ 2^(n - 1/2) * n^(n-1) * s^(2*n + 1) * log(s)^(n + 1/2) / (sqrt(1 + 2*log(s) - 4*log(s)^2) * exp(n) * (1 - 2*log(s))^n), where s = 1.473428520956658037187728756446912746332041803082... is the root of the equation 2*log(s)*(1 + LambertW(log(s))) = 1. - Vaclav Kotesovec, Feb 17 2023