A390059 E.g.f. A(x) satisfies A(x) = exp( x * (1+x^3) * A(x)^2 ).

1, 1, 5, 49, 753, 15241, 388933, 12002985, 435036353, 18115243633, 852350145381, 44725095326209, 2589659638894705, 164020777724356281, 11280807683361410405, 837266619528216292921, 66702818095001953991937, 5677535517818779917547105, 514206359507980014980735557

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (2*(n-3*k)+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.

E.g.f.: exp( -LambertW(-2*x * (1+x^3))/2 ).

Mathematica

a[n_]:=n!*Sum[(2*(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 26 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, (2*(n-3*k)+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);
(Magma) [Factorial(n) * &+[ (2*(n-3*k)+1)^(n-3*k-1) * Binomial(n-3*k, k) / Factorial(n-3*k) : k in [0..Floor(n/4)]] : n in [0..25] ]; // Vincenzo Librandi, Oct 26 2025

Crossrefs

Cf. A362773, A390058.

Cf. A376578.

Sequence in context: A383991 A301386 A192557 * A390302 A357335 A290755

Adjacent sequences: A390056 A390057 A390058 * A390060 A390061 A390062

Keyword

nonn

Author

Seiichi Manyama, Oct 23 2025

Status

approved