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A225615
Partial sums of the binomial coefficients C(5*n,n).
3
1, 6, 51, 506, 5351, 58481, 652256, 7376776, 84281461, 970444596, 11242722766, 130896288616, 1530255133591, 17951328648871, 211205085558031, 2491217772274111, 29449438902782636, 348806466779875961, 4138454609488474736, 49176494325141603881
OFFSET
0,2
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
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)).
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}]
PROG
(PARI) for(n=0, 50, print1(sum(k=0, n, binomial(5*k, k)), ", ")) \\ G. C. Greubel, Apr 01 2017
CROSSREFS
Cf. A006134 (p=2), A188675 (p=3), A225612 (p=4).
Sequence in context: A363112 A180901 A199685 * A002295 A369507 A215159
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Aug 06 2013
STATUS
approved