A225615 - OEIS

%I #30 Apr 02 2017 04:50:41

%S 1,6,51,506,5351,58481,652256,7376776,84281461,970444596,11242722766,

%T 130896288616,1530255133591,17951328648871,211205085558031,

%U 2491217772274111,29449438902782636,348806466779875961,4138454609488474736,49176494325141603881

%N Partial sums of the binomial coefficients C(5*n,n).

%C 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.

%H G. C. Greubel, <a href="/A225615/b225615.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) = (3381*n^4 - 6634*n^3 + 4551*n^2 - 1274*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)/(2869*sqrt(Pi*n)*2^(8*n+3/2)).

%p A225615:=n->add(binomial(5*k,k), k=0..n): seq(A225615(n), n=0..30); # _Wesley Ivan Hurt_, Apr 01 2017

%t Table[Sum[Binomial[5*k, 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 01 2017

%Y Cf. A006134 (p=2), A188675 (p=3), A225612 (p=4).

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Aug 06 2013