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%I #20 Apr 04 2017 03:16:04
%S 1,3,25,195,1625,13879,120717,1063323,9454977,84688303,762972225,
%T 6906366907,62762167561,572251392039,5232480571761,47961608620959,
%U 440565328458621,4054586252967027,37377503512616413,345083448151227987,3190232694060946333,29529002023029712547
%N Alternate partial sums of the binomial coefficients C(4*n,n).
%C 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.
%H G. C. Greubel, <a href="/A227995/b227995.txt">Table of n, a(n) for n = 0..1000</a>
%F 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).
%F a(n) ~ 2^(8*n+17/2)/(283*sqrt(Pi*n)*3^(3*n+1/2)).
%t Table[Sum[Binomial[4*k, k]*(-1)^(n-k), {k, 0, n}], {n, 0, 20}]
%o (PARI) for(n=0,50, print1(sum(k=0,n, binomial(4*k,k)), ", ")) \\ _G. C. Greubel_, Apr 03 2017
%Y Cf. A054108(n-1) (p=2), A188676 (p=3), A227996 (p=5).
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Aug 06 2013
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