A390014 - OEIS

%I #18 Oct 25 2025 09:36:23

%S 1,1,3,16,101,936,11047,157144,2666169,52078672,1150141931,

%T 28347264144,770603470837,22903367650216,738840293877519,

%U 25708877220859816,959844376755705713,38274296147683826976,1623490328244862405459,72992587093309269078688,3467473188638256590907501

%N E.g.f. A(x) satisfies A(x) = exp( x * (1-x^3) * A(x) ).

%H Vincenzo Librandi, <a href="/A390014/b390014.txt">Table of n, a(n) for n = 0..390</a>

%F a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * (n-3*k+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.

%F E.g.f.: exp( -LambertW(-x * (1-x^3)) ).

%t a[n_]:=n!*Sum[(-1)^k*(n-3*k+1)^(n-3*k-1)*Binomial[n-3*k,k]/(n-3*k)!,{k,0,Floor[n/4]}]; Table[a[n],{n,0,25}] (* _Vincenzo Librandi_, Oct 24 2025 *)

%o (PARI) a(n) = n!*sum(k=0, n\4, (-1)^k*(n-3*k+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);

%o (Magma) a := func< n | Factorial(n) * &+[ (-1)^k * (n-3*k + 1)^(n-3*k - 1) * Binomial(n-3*k, k) / Factorial(n-3*k) : k in [0..Floor(n/4)] ] >;

%o [a(n) : n in [0..25]]; // _Vincenzo Librandi_, Oct 24 2025

%Y Cf. A389104, A390013.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 22 2025