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

%I A342107 #23 Dec 23 2024 02:59:58
%S A342107 1,25,2545,372145,63435145,11796180169,2320539673225,474838887231625,
%T A342107 100035931337622625,21552788197602942625,4726913659271173170145,
%U A342107 1051798742538350304851425,236861100204680963085573025
%N A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.
%C A342107 Partial sums of A008977.
%C A342107 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
%F A342107 a(n) ~ 2^(8*n + 15/2) / (255 * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Feb 28 2021
%F A342107 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
%p A342107 A342107 := proc(n)
%p A342107     add((4*k)!/k!^4,k=0..n) ;
%p A342107 end proc:
%p A342107 seq(A342107(n),n=0..70) ; # _R. J. Mathar_, Dec 04 2023
%t A342107 Table[Sum[(4*k)!/k!^4, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 28 2021 *)
%o A342107 (PARI) a(n) = sum(k=0, n, (4*k)!/k!^4);
%Y A342107 Cf. A006134, A008977, A188441, A221177.
%K A342107 nonn,easy
%O A342107 0,2
%A A342107 _Seiichi Manyama_, Feb 28 2021