A227996 - OEIS

%I #20 Apr 04 2017 03:16:14

%S 1,4,41,414,4431,48699,545076,6179444,70725241,815437894,9456840276,

%T 110196725574,1289162119401,15131911395879,178121845513281,

%U 2101890841202799,24856330289305726,294500697587787599,3495147445120811176,41542892270532317969,494440478133277365001

%N Alternate partial sums of the binomial coefficients C(5*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="/A227996/b227996.txt">Table of n, a(n) for n = 0..900</a>

%F Recurrence: 8*n*(2*n-1)*(4*n-3)*(4*n-1)*a(n) = (2869*n^4 - 5866*n^3 + 4199*n^2 - 1226*n + 120)*a(n-1) + 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-2).

%F a(n) ~ 5^(5*n+11/2)/(3381*sqrt(Pi*n)*2^(8*n+3/2)).

%t Table[Sum[Binomial[5*k, k]*(-1)^(n-k), {k, 0, n}], {n, 0, 20}]

%o (PARI) for(n=0,50, print1(sum(k=0,n, binomial(5*k,k)), ", ")) \\ _G. C. Greubel_, Apr 03 2017

%Y Cf. A054108(n-1) (p=2), A188676 (p=3), A227995 (p=4).

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Aug 06 2013