A291449 Numerators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n} (-1)^(n-k)* Stirling2(n, k)*k!*x^k.

1, 1, 13, 1, 43, -61, 728877, 81739, -1779449713, -2112052153, 730622680308569, 113221320488699, -3660430816956396309, -3021604582205161, 21842539561810574341396283, 66747470298418575790593659, -124586733960451680357554181608419, -28471605423890788373026535240299

Offset: 0

Author

Peter Luschny, Aug 24 2017

Keywords

• sign
• frac

Comments

Consider a family of integrals I(m, n) = Integral_{x=0..1} P(n, x)^m with P(n, x) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*x^k. I(1, n) are the Bernoulli numbers A164555/A027642, I(2, n) are the Bernoulli median numbers A212196/A181131, I(3, n) are the numbers A291449/A291450. The coefficients of the polynomials P(n, x)^m are for m = 1 A290694/A290695, for m = 2 A291447/A291448. (See A290694 for further comments.)

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Programs

Maple

# Function BG_row is defined in A290694.
seq(BG_row(3, n, "num", "val"), n=0..17);

Mathematica

P[n_, x_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k!*x^k, {k, 0, n}];
a[n_] := Integrate[P[n, x]^3, {x, 0, 1}] // Numerator;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 15 2019 *)