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

0,2

Links

Table of n, a(n) for n=0..19.

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

Recurrence (of order 2): 3*(n-1)*n*(3*n - 2)*(3*n - 1)*(186*n^2 - 527*n + 370)*a(n) = - 72*(n-1)*(7254*n^5 - 31434*n^4 + 50576*n^3 - 37284*n^2 + 12457*n - 1485)*a(n-1) + 80*(5*n - 8)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(186*n^2 - 155*n + 29)*a(n-2). - Vaclav Kotesovec, Nov 09 2025

Prog

(PARI) a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(2*n-k, n-k));

Crossrefs

Cf. A386812, A386896, A386897, A386898.

Sequence in context: A214385 A121161 A360200 * A098269 A010565 A391555

Adjacent sequences: A386892 A386893 A386894 * A386896 A386897 A386898

Keyword

nonn

Author

Seiichi Manyama, Aug 07 2025

Status

approved