A227995 Alternate partial sums of the binomial coefficients C(4*n,n).

1, 3, 25, 195, 1625, 13879, 120717, 1063323, 9454977, 84688303, 762972225, 6906366907, 62762167561, 572251392039, 5232480571761, 47961608620959, 440565328458621, 4054586252967027, 37377503512616413, 345083448151227987, 3190232694060946333, 29529002023029712547

Offset: 0

Author

Vaclav Kotesovec, Aug 06 2013

Keywords

• nonn
• easy

Comments

Generally (for p>1), alternate partial sums of the binomial coefficients C(p*n,n) is 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. A054108(n-1) (p=2), A188676 (p=3), A227996 (p=5).

Programs

Mathematica

Table[Sum[Binomial[4*k, k]*(-1)^(n-k), {k, 0, n}], {n, 0, 20}]

PARI

for(n=0,50, print1(sum(k=0,n, binomial(4*k,k)), ", ")) \\ G. C. Greubel, Apr 03 2017

Formula

Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (229*n^3 - 357*n^2 + 170*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)/(283*sqrt(Pi*n)*3^(3*n+1/2)).