A384017 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^5.

1, 5, 100, 3510, 177370, 11732175, 960453825, 93791830160, 10644367637490, 1376936603007075, 200002385378370350, 32233130183113838550, 5708169533474858008905, 1101836121788665346133960, 230256048227047074266497400, 51791322674249971562728368000

Offset: 0

Author

Seiichi Manyama, May 18 2025

Keywords

• nonn

Comments

From Vaclav Kotesovec, May 19 2025: (Start)

In general, for m > 1, [x^n] Product_{k=0..n} (1 + k*x)^m ~ m^(m*(n + 1/2)) * w^(m*n + (m+1)/2) * n^(n - 1/2) / (sqrt(2*Pi*(w-1)) * exp(n) * (m-1)^((m-1)*n + (m+1)/2) * (m*w-m+1)^n), where w = -LambertW(-1,-(m-1)*exp(-(m-1)/m)/m).

The general formula is valid even for m=n, where after modifications we get the formula for A351507. (End)

Crossrefs

Cf. A000142 (m=1), A129256 (m=2), A384012 (m=3), A384031 (m=4), A351507 (m=n).

Programs

Mathematica

Table[SeriesCoefficient[Product[(1+k*x)^5, {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 19 2025 *)

PARI

a(n) = polcoef(prod(k=1, n, 1+k*x)^5, n);

Formula

a(n) = Sum_{0<=i, j, k, l, m<=n and i+j+k+l+m=4*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1) * Stirling1(n+1,m+1)|.

a(n) ~ 5^(5*n + 5/2) * w^(5*n+3) * n^(n - 1/2) / (2^(8*n + 13/2) * sqrt(Pi*(w-1)) * exp(n) * (5*w-4)^n), where w = -LambertW(-1,-4*exp(-4/5)/5) = 1.2308422097842590367678406745433500325966... - Vaclav Kotesovec, May 19 2025