A227996 Alternate partial sums of the binomial coefficients C(5*n,n).

1, 4, 41, 414, 4431, 48699, 545076, 6179444, 70725241, 815437894, 9456840276, 110196725574, 1289162119401, 15131911395879, 178121845513281, 2101890841202799, 24856330289305726, 294500697587787599, 3495147445120811176, 41542892270532317969, 494440478133277365001

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..900

Crossrefs

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

Programs

Mathematica

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

PARI

for(n=0,50, print1(sum(k=0,n, binomial(5*k,k)), ", ")) \\ G. C. Greubel, Apr 03 2017

Formula

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).

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