A390416 a(n) = Sum_{k=0..n} binomial(4*n+4*k+1,n-k).

1, 6, 50, 444, 4040, 37219, 345411, 3220935, 30134735, 282618136, 2655353228, 24983716550, 235330304876, 2218665368404, 20932872176739, 197621850423870, 1866671633119329, 17639815328729024, 166757771009184464, 1576964222516449178, 14917050056138790196

Offset

0,2

Links

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

Formula

G.f.: g/((1-4*x*g^3) * (1-x*g^8)) where g = 1+x*g^4 is the g.f. of A002293.

a(n) = binomial(1+4*n, n)*hypergeom([1, -n, 1/2+n, 3/4+n, 1+n, 5/4+n], [(2+3*n)/5, 3*(1+n)/5, (4+3*n)/5, 1+3*n/5, (6+3*n)/5], -2^8/5^5). - Stefano Spezia, Nov 06 2025

Mathematica

Table[Sum[Binomial[4*n+4*k+1, n-k], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Nov 06 2025 *)
a[n_]:=Binomial[1+4*n, n]*HypergeometricPFQ[{1, -n, 1/2+n, 3/4+n, 1+n, 5/4+n}, {(2+3*n)/5, 3*(1+n)/5, (4+3*n)/5, 1+3*n/5, (6+3*n)/5}, -2^8/5^5]; Array[a, 21, 0] (* Stefano Spezia, Nov 06 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(4*n+4*k+1, n-k));

Crossrefs

Cf. A257633, A386811, A390333, A390411, A390412, A390413, A390414, A390415.

Cf. A002293.

Sequence in context: A027330 A090409 A212233 * A180910 A199680 A039742

Adjacent sequences: A390413 A390414 A390415 * A390417 A390418 A390419

Keyword

nonn

Author

Seiichi Manyama, Nov 04 2025

Status

approved