A386897 a(n) = 4^n * binomial(5*n/2,n).
1, 10, 160, 2860, 53760, 1040060, 20500480, 409404600, 8255569920, 167718033340, 3427543285760, 70384350760360, 1451115518361600, 30018413447053080, 622759359440486400, 12951795276279787760, 269947721071617638400, 5637113741080428839100
Offset: 0
Links
- Paolo Xausa, Table of n, a(n) for n = 0..700
Programs
-
Mathematica
Table[Sum[2^k *(-1)^(n-k)*Binomial[5*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *) A386897[n_] := 4^n*Binomial[5*n/2, n]; Array[A386897, 20, 0] (* Paolo Xausa, Aug 26 2025 *) -
PARI
a(n) = 4^n*binomial(5*n/2, n);
Formula
a(n) == 0 (mod 10) for n > 0.
a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k) * binomial(2*n-k,n-k).
a(n) = [x^n] 1/(1-4*x)^(3*n/2+1).
a(n) = [x^n] (1+4*x)^(5*n/2).
a(n) ~ 2^(n - 1/2) * 5^((5*n+1)/2) / (sqrt(Pi*n) * 3^((3*n+1)/2)). - Vaclav Kotesovec, Aug 07 2025
D-finite with recurrence 3*n*(n-1)*(3*n-4) *(3*n-2)*a(n) -20*(5*n-4) *(5*n-8)*(5*n-2) *(5*n-6)*a(n-2)=0. - R. J. Mathar, Aug 21 2025
O.g.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (12500*x^2)/27) + 10*x*hypergeom([7/10, 9/10, 11/10, 13/10], [5/6, 7/6, 3/2], (12500*x^2)/27). - Karol A. Penson, Aug 26 2025