A386614 a(n) = Sum_{k=0..n-1} binomial(5*k+1,k) * binomial(5*n-5*k,n-k-1).

0, 1, 16, 220, 2880, 36850, 465536, 5834852, 72744640, 903525715, 11191199200, 138323478980, 1706860996096, 21034268215120, 258934785258240, 3184696786012500, 39140208951032960, 480734044749851305, 5901368553964031600, 72410017973538837880, 888114187330722044800, 10888921795007470528060

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: g^3 * (g-1)/(5-4*g)^2 where g=1+x*g^5.

G.f.: g/((1-g)^2 * (1-5*g)^2) where g*(1-g)^4 = x.

a(n) = Sum_{k=0..n-1} binomial(5*k+1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.

a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n+2,k).

a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k+2,k).

D-finite with recurrence +35651584*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) +8192*(56348704*n^4-268019168*n^3+418502324*n^2-264019618*n+57303885)*a(n-1) +160*(-65524820000*n^4+314102050000*n^3-463341186250*n^2+159732814775*n+76118151939)*a(n-2) +62500*(660806875*n^4-1813661250*n^3-5080986250*n^2+20705993100*n-17279228304)*a(n-3) +308935546875*(5*n-11)*(5*n-14)*(5*n-13)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

Recurrence (of order 2): 2048*n*(2*n + 1)*(4*n - 1)*(4*n + 1)*(200*n^2 - 317*n + 122)*a(n) = 40*(8000000*n^6 - 14680000*n^5 + 6140000*n^4 + 1531250*n^3 - 1042375*n^2 + 30854*n + 13440)*a(n-1) - 15625*(5*n - 7)*(5*n - 4)*(5*n - 3)*(5*n - 1)*(200*n^2 + 83*n + 5)*a(n-2). - Vaclav Kotesovec, Nov 09 2025

Prog

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

Crossrefs

Cf. A079678, A386367, A386566, A386613.

Cf. A006419, A386612, A386616, A386617.

Sequence in context: A130606 A193380 A076073 * A092657 A078266 A093746

Adjacent sequences: A386611 A386612 A386613 * A386615 A386616 A386617

Keyword

nonn

Author

Seiichi Manyama, Jul 27 2025

Status

approved