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

0,2

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

Formula

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

From Seiichi Manyama, Sep 13 2025: (Start)

a(n) = [x^n] (1+x)^n/(1-x)^(5*n+1).

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(5*n,k).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(5*n+k,n). (End)

a(n) = [x^n] ( (1+2*x) * (1+x)^5 )^n. - Seiichi Manyama, Sep 20 2025

Mathematica

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

Prog

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

Crossrefs

Column k=5 of A387934.

Cf. A001850, A114496, A156886, A156887.

Cf. A243668, A341491.

Sequence in context: A249448 A069661 A069662 * A371386 A295543 A062747

Adjacent sequences: A359643 A359644 A359645 * A359647 A359648 A359649

Keyword

nonn

Author

Vaclav Kotesovec, Jan 09 2023

Status

approved