A225612 Partial sums of the binomial coefficients C(4*n,n).

1, 5, 33, 253, 2073, 17577, 152173, 1336213, 11854513, 105997793, 953658321, 8622997453, 78291531921, 713305091521, 6518037055321, 59712126248041, 548239063327621, 5043390644753269, 46475480410336709, 428936432074181109, 3964252574286355429

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..1000

Crossrefs

Cf. A006134 (p=2), A188675 (p=3), A225615 (p=5).

Programs

Maple

A225612:=n->add(binomial(4*k,k), k=0..n): seq(A225612(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2017

Mathematica

Table[Sum[Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[4n,n],{n,0,20}]] (* Harvey P. Dale, Feb 01 2015 *)

PARI

for(n=0,50, print1(sum(k=0,n, binomial(4*k,k)), ", ")) \\ G. C. Greubel, Apr 01 2017

Formula

Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (283*n^3 - 411*n^2 + 182*n - 24)*a(n-1) - 8*(2*n-1)*(4*n-3)*(4*n-1)*a(n-2).

a(n) ~ 2^(8*n+17/2)/(229*sqrt(Pi*n)*3^(3*n+1/2)).