A390500 a(n) = Sum_{k=0..n} (-1)^k * (3*k+1) * binomial(4*n-k+1,n-k)/(4*n-k+1).

1, 0, 1, 6, 40, 284, 2110, 16210, 127756, 1027320, 8395793, 69534450, 582337636, 4923270360, 41962170620, 360182765930, 3110783981180, 27013818430976, 235727430754324, 2065960277673720, 18177491588096980, 160503722749106380, 1421792661854503270

Offset

0,4

Links

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

Formula

G.f.: g/(1 + x*g^3), where g = 1+x*g^4 is the g.f. of A002293.

From Seiichi Manyama, Nov 29 2025: (Start)

G.f.: 1/(1 - x^2*g^6), where g = 1+x*g^4 is the g.f. of A002293.

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

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

Mathematica

Table[Sum[(-1)^k*(3*k+1)*Binomial[4*n-k+1, n-k]/(4*n-k+1), {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Nov 08 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*(3*k+1)*binomial(4*n-k+1, n-k)/(4*n-k+1));
(Magma) [&+[(-1)^k*(3*k+1)*Binomial(4*n-k+1, n-k)/(4*n-k+1): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Nov 08 2025

Crossrefs

Cf. A000957, A089354, A390744.

Cf. A002293, A391084.

Sequence in context: A178397 A090041 A069720 * A005037 A081337 A394232

Adjacent sequences: A390497 A390498 A390499 * A390501 A390502 A390503

Keyword

nonn

Author

Seiichi Manyama, Nov 08 2025

Status

approved