A385632 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n+1,k).

1, 8, 81, 872, 9669, 109128, 1246419, 14359304, 166512285, 1940885504, 22717923586, 266833238328, 3143237113479, 37119019790016, 439290932937672, 5208668386199112, 61861932606093901, 735804601177846968, 8763478151940329859, 104498114621004830160, 1247410783999193335434

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n+1)).

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

a(n) = 3^(5*n+1)*2^(-4*n-1) - binomial(5*n+1, n)*(hypergeom([1, -1-4*n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k). - Seiichi Manyama, Aug 07 2025

G.f.: g^2/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 14 2025

From Seiichi Manyama, Aug 16 2025: (Start)

G.f.: 1/(1 - x*g^3*(15-7*g)) where g = 1+x*g^5 is the g.f. of A002294.

G.f.: B(x)^2/(1 + 2*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)

Prog

(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(5*n+1, k));

Crossrefs

Cf. A006256, A141223, A385605.

Cf. A001449, A079589, A079678, A371753, A386812.

Cf. A386699.

Sequence in context: A324016 A068617 A207994 * A210127 A007778 A065440

Adjacent sequences: A385629 A385630 A385631 * A385633 A385634 A385635

Keyword

nonn

Author

Seiichi Manyama, Aug 03 2025

Status

approved