A387013 - OEIS

%I #54 Oct 26 2025 12:09:56

%S 1,1,3,28,269,3336,53287,1008064,22199193,559954432,15899946731,

%T 502178632704,17470278315877,663858654275584,27361365974132751,

%U 1215869626302939136,57952329108501627953,2949335923883992743936,159631304307995442321235,9156282245864958498439168

%N E.g.f. A(x) satisfies A(x) = exp(x * (1+x^2)^2 * A(x)).

%H Vincenzo Librandi, <a href="/A387013/b387013.txt">Table of n, a(n) for n = 0..345</a>

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

%F E.g.f.: exp( -LambertW(-x*(1+x^2)^2) ).

%t a[n_]:=n!*Sum[(n-2*k+1)^(n-2*k-1)*Binomial[2*(n-2*k),k]/(n-2*k)!,{k,0,Floor[n/2]}];Table[a[n],{n,0,25}] (* _Vincenzo Librandi_, Oct 26 2025 *)

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

%o (Magma) [Factorial(n) * &+[(n-2*k+1)^(n-2*k-1) * Binomial(2*(n-2*k), k) / Factorial(n-2*k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // _Vincenzo Librandi_, Oct 26 2025

%Y Cf. A376577, A389986.

%Y Cf. A362772, A389987.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 21 2025