A387971 E.g.f. A(x) satisfies A(x) = exp( x^2*A(x)^2 * (1+x*A(x)) ) * (1+x*A(x)).

1, 1, 4, 36, 444, 7200, 147720, 3650640, 105532560, 3498526080, 130876179840, 5454135751680, 250608029474880, 12587948008796160, 686247657362467200, 40356310997421619200, 2546556142205678803200, 171636384521080339660800, 12306251107076561924889600

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)) / (1+x) ).

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

Mathematica

Table[n!*Sum[(n+1)^(k-1)*Binomial[n+k+1, n-2*k]/k!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Oct 27 2025 *)
m=25; a[0]=1; trunc[n_]:=Sum[a[k] x^k/Factorial[k], {k, 0, n-1}]; Do[a[n] =Factorial[n] Coefficient[Normal@Series[Exp[x^2 trunc[n]^2 (1+x trunc[n])] (1+x trunc[n]), {x, 0, n}], x, n], {n, 1, m}]; Table[a[n], {n, 0, m}] (* Vincenzo Librandi, Oct 29 2025 *)

Prog

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

Crossrefs

Cf. A387972, A387973, A387974.

Cf. A181734.

Sequence in context: A374857 A136224 A321963 * A307903 A213596 A383774

Adjacent sequences: A387968 A387969 A387970 * A387972 A387973 A387974

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2025

Status

approved