A390548 a(n) = Sum_{k=0..n} (-1)^k * binomial(5*n-k+2,n-k).

1, 6, 56, 574, 6157, 67826, 760384, 8630926, 98877509, 1140897484, 13239577408, 154356072202, 1806569021831, 21213157190064, 249787786119264, 2948416161538734, 34875908334254461, 413306724373243634, 4906128830898921976, 58324188981968175740, 694283772469428548282

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Formula

G.f.: g^2/((1-5*x*g^4) * (1+x*g^4)) where g = 1+x*g^5 is the g.f. of A002294.

a(n) = Sum_{k=0..n} (-2)^k * binomial(5*n+3,n-k).

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

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

From Vaclav Kotesovec, Nov 11 2025: (Start)

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+k+2,k).

a(n) ~ 5^(5*n + 7/2) / (3 * sqrt(Pi*n) * 2^(8*n + 13/2)). (End)

Mathematica

Table[Sum[(-2)^k*Binomial[5*n+3, n-k], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Nov 10 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(5*n-k+2, n-k));
(Magma) [&+[(-2)^k*Binomial(5*n+3, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Nov 10 2025

Crossrefs

Cf. A014301, A199033, A390547.

Cf. A371753, A387756, A390590, A390593.

Cf. A002294.

Sequence in context: A183615 A181589 A093142 * A092655 A099140 A371739

Adjacent sequences: A390545 A390546 A390547 * A390549 A390550 A390551

Keyword

nonn

Author

Seiichi Manyama, Nov 10 2025

Status

approved