A392978 Expansion of (1/x) * Series_Reversion( x * ((1-x^3)^2 - x) ).

1, 1, 2, 7, 26, 98, 387, 1590, 6699, 28754, 125359, 553696, 2472267, 11140492, 50599174, 231401382, 1064646228, 4924458114, 22886125061, 106815213813, 500447918795, 2352844049280, 11096870017710, 52488362170335, 248930058920517, 1183456657164537, 5639068382464958

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f. A(x) satisfies A(x) = 1/((1 - (x*A(x))^3)^2 - x*A(x)).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n-3*k,n) * binomial(4*n-5*k+1,k).

Mathematica

Table[1/(n+1)*Sum[Binomial[2*n-3*k, n]*Binomial[4*n-5*k+1, k], {k, 0, Floor[n/3]}], {n, 0, 24}] (* Vincenzo Librandi, Jan 31 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(2*n-3*k, n)*binomial(4*n-5*k+1, k))/(n+1);
(Magma) [1/(n+1) * &+[Binomial(2*n-3*k, n)* Binomial(4*n-5*k+1, k) : k in [0..Floor(n/3)]] : n in [0..26] ]; // Vincenzo Librandi, Jan 31 2026

Crossrefs

Cf. A049140, A370840, A392979.

Sequence in context: A126223 A369231 A369489 * A389540 A273320 A114121

Adjacent sequences: A392975 A392976 A392977 * A392979 A392980 A392981

Keyword

nonn

Author

Seiichi Manyama, Jan 29 2026

Status

approved