A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.

1, 25, 2545, 372145, 63435145, 11796180169, 2320539673225, 474838887231625, 100035931337622625, 21552788197602942625, 4726913659271173170145, 1051798742538350304851425, 236861100204680963085573025

Offset: 0

Author

Seiichi Manyama, Feb 28 2021

Keywords

• nonn
• easy

Comments

Partial sums of A008977.

In general, for m > 1, Sum_{k=0..n} (m*k)!/k!^m ~ m^(m*n + m + 1/2) / ((m^m - 1) * (2*Pi*n)^((m-1)/2)). - Vaclav Kotesovec, Feb 28 2021

Crossrefs

Cf. A006134, A008977, A188441, A221177.

Programs

Maple

A342107 := proc(n)
    add((4*k)!/k!^4,k=0..n) ;
end proc:
seq(A342107(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

Mathematica

Table[Sum[(4*k)!/k!^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2021 *)

PARI

a(n) = sum(k=0, n, (4*k)!/k!^4);

Formula

a(n) ~ 2^(8*n + 15/2) / (255 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 28 2021

D-finite with recurrence n^3*a(n) +(-257*n^3+384*n^2-176*n+24)*a(n-1) +8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Dec 04 2023