A377859 - OEIS

%I #10 Nov 10 2024 03:34:55

%S 1,0,1,2,21,144,1765,21552,340137,5845760,116495721,2550320640,

%T 62023290109,1642735460352,47321500546125,1469008742856704,

%U 48962556079079505,1742660440701861888,65993849612007279697,2648999558505185280000,112360563741545020804581

%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} (-1)^k * (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(phi*n + 1/phi)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Nov 10 2024

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

%Y Cf. A377860, A377861.

%Y Cf. A377831.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Nov 09 2024