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

1, 7, 89, 1273, 19181, 297662, 4707971, 75459496, 1221388525, 19919031781, 326797222834, 5387618403526, 89178832899887, 1481143718244912, 24671054686539336, 411966653603163008, 6894167059382069485, 115593504497163747167, 1941434442814233362939, 32656575110841643234631

Offset: 0

Author

Vaclav Kotesovec, Jan 09 2023

Keywords

• nonn

Comments

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.

Equivalently, d = (3 + m + sqrt(1 + m*(6 + m))) * (1 + 3*m + sqrt(1 + m*(6 + m)))^m / (2^(2*m + 1) * m^m).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A001850, A114496, A156886, A156887.

Programs

Mathematica

Table[Sum[Binomial[n, k]*Binomial[5*n+k, k], {k, 0, n}], {n, 0, 20}]

PARI

a(n) = sum(k=0, n, binomial(n,k) * binomial(5*n+k,k)) \\ Andrew Howroyd, Jan 09 2023

Formula

a(n) ~ sqrt(3/10 + 23/(20*sqrt(14))) * ((108007 + 28854*sqrt(14))/12500)^n / sqrt(Pi*n).