A349936 Central pentanomial coefficients.

1, 5, 85, 1751, 38165, 856945, 19611175, 454805755, 10651488789, 251345549849, 5966636799745, 142330448514875, 3408895901222375, 81922110160246231, 1974442362935339179, 47705925773278538281, 1155170746105476171285, 28025439409568101909625, 681077893998769910221225

Offset: 0

Author

Stefano Spezia, Dec 06 2021

Keywords

• nonn
• easy

Comments

Largest coefficient of (Sum_{j=0..4} x^j)^(2*n).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Central coefficients in triangle A035343.

Column s = 4 in A349933.

Cf. A005721, A063419, A082758.

Programs

Mathematica

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; Table[T[2n,4n,4],{n,0,18}]

Formula

a(n) = T(2*n, 4*n, 4), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.

a(n) = A035343(2*n, 4*n) = [x^(4*n)] (Sum_{j=0..4} x^j)^(2*n).

From Vaclav Kotesovec, Dec 09 2021: (Start)

Recurrence: 2*n*(2*n - 1)*(3*n - 4)*(4*n - 7)*(4*n - 3)*(4*n - 1)*(6*n - 13)*(6*n - 7)*a(n) = 3*(4*n - 7)*(6*n - 13)*(10584*n^6 - 47628*n^5 + 84190*n^4 - 73965*n^3 + 33531*n^2 - 7272*n + 570)*a(n-1) - 75*(n-1)*(2*n - 3)*(4*n - 5)*(6*n - 1)*(504*n^4 - 2520*n^3 + 4160*n^2 - 2525*n + 476)*a(n-2) + 625*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(3*n - 1)*(4*n - 3)*(6*n - 7)*(6*n - 1)*a(n-3).

a(n) ~ 25^n / sqrt(8*Pi*n). (End)