A394487 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + 4*x)^(1/4) ).

1, 1, -1, 0, 21, -240, 1755, 0, -329175, 7983360, -109643625, 0, 57032531325, -2117187072000, 42706082293875, 0, -43517817539571375, 2176095685091328000, -57912650546666540625, 0, 97531915080599667793125, -6133978517135435366400000, 202771821952414878685396875, 0

Offset

0,5

Links

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + 4*x*A(x))^(1/4).

a(n) = 4^n * n! * binomial((n+1)/4,n)/(n+1).

a(4*n+3) = 0.

Mathematica

a[n_]:=4^n*n!*Binomial[( n+1)/4, n]/(n+1); Table[a[n], {n, 0, 20}] (* Vincenzo Librandi, Apr 01 2026 *)

Prog

(PARI) a(n) = 4^n*n!*binomial((n+1)/4, n)/(n+1);
(Magma) [1] cat [4^n*Factorial(n)*&*[((n+1)/4-k): k in [0..n-1]]/Factorial(n)/(n+1): n in [1..20] ]; // Vincenzo Librandi, Apr 01 2026

Crossrefs

Cf. A000142, A001761, A282626, A385369, A385440, A393169.

Sequence in context: A316426 A317428 A089937 * A175843 A198974 A271792

Adjacent sequences: A394484 A394485 A394486 * A394488 A394489 A394490

Keyword

sign,easy

Author

Seiichi Manyama, Mar 22 2026

Status

approved