A386895 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(2*n-k,n-k).
1, 8, 94, 1220, 16590, 231808, 3297154, 47490696, 690461070, 10111370720, 148929775544, 2203898519732, 32741261744802, 488010179737920, 7294326822378060, 109294796958693520, 1641111255497600910, 24688289062391137056, 372020649062760239080, 5614219481885985162960
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(2*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(3*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(4*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-5*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
D-finite with recurrence +135*n*(n-1)*(3*n-1)*(3*n-2)*a(n) +3*(n-1)*(104049*n^3 -434754*n^2 +745789*n -439424)*a(n-1) +36*(517211*n^4 -4353801*n^3 +13137926*n^2 -17477238*n +8846684)*a(n-2) +16*(-11442763*n^4 +46270475*n^3 +85309279*n^2 -584322689*n +652846590)*a(n-3) -4585920*(5*n-16) *(5*n-14) *(5*n-18)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 21 2025