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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
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)).
Comments