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A359646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+k,k).

%I A359646 #10 Jan 09 2023 21:30:47
%S A359646 1,7,89,1273,19181,297662,4707971,75459496,1221388525,19919031781,
%T A359646 326797222834,5387618403526,89178832899887,1481143718244912,
%U A359646 24671054686539336,411966653603163008,6894167059382069485,115593504497163747167,1941434442814233362939,32656575110841643234631
%N A359646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+k,k).
%C A359646 In general, for m>0, Sum_{k=0..n} binomial(n,k) * binomial(m*n+k,k) ~ (m+c) / sqrt(2*Pi*c*m * (m*(2-c)+c)*n) * d^n, where d = (m+c)^(m+c) / ((1-c)^(1-c) * c^(2*c) * m^m) and c = (sqrt(m^2 + 6*m + 1) + 1 - m)/4.
%C A359646 Equivalently, d = (3 + m + sqrt(1 + m*(6 + m))) * (1 + 3*m + sqrt(1 + m*(6 + m)))^m / (2^(2*m + 1) * m^m).
%H A359646 Andrew Howroyd, <a href="/A359646/b359646.txt">Table of n, a(n) for n = 0..500</a>
%F A359646 a(n) ~ sqrt(3/10 + 23/(20*sqrt(14))) * ((108007 + 28854*sqrt(14))/12500)^n / sqrt(Pi*n).
%t A359646 Table[Sum[Binomial[n, k]*Binomial[5*n+k, k], {k, 0, n}], {n, 0, 20}]
%o A359646 (PARI) a(n) = sum(k=0, n, binomial(n,k) * binomial(5*n+k,k)) \\ _Andrew Howroyd_, Jan 09 2023
%Y A359646 Cf. A001850, A114496, A156886, A156887.
%K A359646 nonn
%O A359646 0,2
%A A359646 _Vaclav Kotesovec_, Jan 09 2023