A390013 - OEIS

%I #20 Oct 23 2025 21:05:38

%S 1,1,3,10,53,336,2887,30304,395817,6102784,109219211,2206139904,

%T 49543830877,1220029069312,32645494819407,942105767366656,

%U 29155253491158353,962920464001990656,33804330124398965011,1257010041289297494016,49356978868692732793221

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

%H Vincenzo Librandi, <a href="/A390013/b390013.txt">Table of n, a(n) for n = 0..400</a>

%F a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (n-2*k+1)^(n-2*k-1) * binomial(n-2*k,k)/(n-2*k)!.

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

%t a[n_]:=n!*Sum[(-1)^k*(n-2*k+1)^(n-2*k-1)*Binomial[n-2*k,k]/(n-2*k)!,{k,0,Floor[n/3]}]; Table[a[n],{n,0,25}] (* _Vincenzo Librandi_, Oct 23 2025 *)

%o (PARI) a(n) = n!*sum(k=0, n\3, (-1)^k*(n-2*k+1)^(n-2*k-1)*binomial(n-2*k, k)/(n-2*k)!);

%o (Magma) a := func< n | Factorial(n) * &+[ (-1)^k * (n-2*k + 1)^(n-2*k - 1) * Binomial(n-2*k, k) / Factorial(n-2*k) : k in [0..Floor(n/3)] ] >;

%o [ a(n) : n in [0..25] ]; // _Vincenzo Librandi_, Oct 23 2025

%Y Cf. A389104, A390014.

%Y Cf. A376577.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 22 2025