A225615 Partial sums of the binomial coefficients C(5*n,n).

1, 6, 51, 506, 5351, 58481, 652256, 7376776, 84281461, 970444596, 11242722766, 130896288616, 1530255133591, 17951328648871, 211205085558031, 2491217772274111, 29449438902782636, 348806466779875961, 4138454609488474736, 49176494325141603881

Offset: 0

Author

Vaclav Kotesovec, Aug 06 2013

Keywords

• nonn
• easy

Comments

Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..900

Crossrefs

Cf. A006134 (p=2), A188675 (p=3), A225612 (p=4).

Programs

Maple

A225615:=n->add(binomial(5*k,k), k=0..n): seq(A225615(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2017

Mathematica

Table[Sum[Binomial[5*k, k], {k, 0, n}], {n, 0, 20}]

PARI

for(n=0,50, print1(sum(k=0,n, binomial(5*k,k)), ", ")) \\ G. C. Greubel, Apr 01 2017

Formula

Recurrence: 8*n*(2*n-1)*(4*n-3)*(4*n-1)*a(n) = (3381*n^4 - 6634*n^3 + 4551*n^2 - 1274*n + 120)*a(n-1) - 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-2).

a(n) ~ 5^(5*n+11/2)/(2869*sqrt(Pi*n)*2^(8*n+3/2)).