A387972 E.g.f. A(x) satisfies A(x) = exp( x^2*A(x)^2 * (1+x*A(x))^2 ) * (1+x*A(x)).

1, 1, 4, 42, 588, 10440, 235920, 6462960, 206776080, 7579232640, 313592005440, 14456132570880, 734714529151680, 40819704806200320, 2461456820010988800, 160110935864341862400, 11175351881103105696000, 833137068038144834764800, 66074345102279325190579200

Offset

0,3

Links

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

Formula

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

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

Mathematica

Table[n!*Sum[(n+1)^(k-1)*Binomial[n+2*k+1, n-2*k]/k!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Oct 27 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (n+1)^(k-1)*binomial(n+2*k+1, n-2*k)/k!);
(Magma) [Factorial(n) * &+[(n+1)^(k-1)* Binomial(n+2*k+1, n-2*k) / Factorial(k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Oct 27 2025

Crossrefs

Cf. A387971, A387973, A387974.

Cf. A199247.

Sequence in context: A156440 A151453 A234507 * A153854 A216080 A137645

Adjacent sequences: A387969 A387970 A387971 * A387973 A387974 A387975

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2025

Status

approved