A384024 a(n) = [x^n] Product_{k=0..n} (1 + (n+k)*x).

1, 3, 26, 342, 5944, 127860, 3272688, 97053936, 3270729600, 123418922400, 5154170774400, 235977273544320, 11752173128586240, 632474276804697600, 36576553723886131200, 2261980049125982976000, 148956705206745595084800, 10406288081667512679321600, 768701832940487804295168000

Offset: 0

Author

Vaclav Kotesovec, May 17 2025

Keywords

• nonn

Crossrefs

Central terms of triangle A165675.

Cf. A000407, A201546, A383869.

Programs

Mathematica

Table[SeriesCoefficient[Product[1 + (n+k)*x, {k, 0, n}], {x, 0, n}], {n, 0, 20}]

PARI

a(n) = sum(k=0, n, (k+1)*n^k*abs(stirling(n+1, k+1, 1))); \\ Seiichi Manyama, May 18 2025

Formula

a(n) ~ n! * log(2) * 4^n * sqrt(n/Pi).

a(n) ~ log(2) * 2^(2*n + 1/2) * n^(n+1) / exp(n).

From Seiichi Manyama, May 18 2025: (Start)

a(n) = A165675(2*n,n).

a(n) = Sum_{k=0..n} (k+1) * n^k * |Stirling1(n+1,k+1)|.

a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-n,k)/(n+1-k).

a(n) = (2*n)!/n! * (1 + n * Sum_{k=1..n} 1/(n+k)). (End)