A371773 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k+1,n-3*k).

1, 3, 10, 36, 134, 507, 1937, 7449, 28783, 111623, 434130, 1692387, 6610292, 25861384, 101319095, 397428091, 1560588454, 6133768656, 24128550045, 94986663925, 374188128311, 1474980414870, 5817387549611, 22955930045826, 90629404431826, 357960414264163

Offset: 0

Author

Seiichi Manyama, Apr 05 2024

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A371774, A371775, A371776.

Cf. A144904, A371758, A371777.

Programs

Mathematica

Table[Sum[Binomial[2n-k+1,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Sep 09 2024 *)

PARI

a(n) = sum(k=0, n\3, binomial(2*n-k+1, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^n).

a(n) = binomial(2*(n-1), n-1)*hypergeom([1, (1-n)/3, (2-n)/3, 1-n/3], [1-n, 3/2-n, n], -27/4). - Stefano Spezia, Apr 06 2024

From Vaclav Kotesovec, Apr 08 2024: (Start)

Recurrence: n*(n^2 - 7)*a(n) = (9*n^3 - 2*n^2 - 79*n + 60)*a(n-1) - 2*(12*n^3 - 5*n^2 - 124*n + 150)*a(n-2) + (17*n^3 - 8*n^2 - 183*n + 240)*a(n-3) - 2*(2*n - 5)*(n^2 + 2*n - 6)*a(n-4).

a(n) ~ 2^(2*n+2) / sqrt(Pi*n). (End)