A377831 - OEIS

%I #10 Nov 09 2024 08:11:57

%S 1,2,13,154,2701,63216,1856569,65711024,2724349401,129552751360,

%T 6952877604421,415770771875328,27416031835737637,1976460653044957184,

%U 154658036515292528625,13055394531339601033216,1182611605875201470044081,114426900236922150187892736

%N Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x) ).

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)).

%F a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-k,n-k)/k!.

%F a(n) ~ phi^(3*n + 3/2) * n^(n-1) / (5^(1/4) * exp((n+1)/phi - 1)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Nov 09 2024

%o (PARI) a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n-k, n-k)/k!);

%Y Cf. A377832, A377833.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 09 2024