A388043 a(n) = Sum_{k=0..n} binomial(3*n-2*k,n-k).

1, 4, 20, 111, 650, 3927, 24207, 151316, 955604, 6082374, 38954135, 250730636, 1620527481, 10510253485, 68368227668, 445864207407, 2914172036874, 19084253300085, 125194638808086, 822556496457206, 5411859876744215, 35650822482539109, 235117170530582284, 1552198373110679575

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/((1-3*x*g^2) * (1-x*g)) where g = 1+x*g^3 is the g.f. of A001764.

a(n) ~ 3^(3*n + 5/2) / (7 * sqrt(Pi*n) * 2^(2*n+1)). - Vaclav Kotesovec, Nov 04 2025

a(n) = Sum_{k=0..n} binomial(n + 2*k, k). - Vaclav Kotesovec, Nov 08 2025

From Seiichi Manyama, Nov 10 2025: (Start)

a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,n-k).

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

Mathematica

Table[Sum[Binomial[n+2*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 04 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(3*n-2*k, n-k));

Crossrefs

Cf. A038736, A045721, A066380, A188675, A390358, A390359.

Cf. A006134, A390407.

Cf. A001764, A390499.

Sequence in context: A026156 A025183 A014523 * A153299 A239643 A081335

Adjacent sequences: A388040 A388041 A388042 * A388044 A388045 A388046

Keyword

nonn

Author

Seiichi Manyama, Nov 02 2025

Status

approved