A192238 Constant term in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.

1, 0, 1, 6, 37, 256, 1999, 17490, 169895, 1816320, 21205745, 268547510, 3667187645, 53722014720, 840455448415, 13985762375970, 246675543859855, 4596826887347200, 90249727067243425, 1861971659969854950, 40274219840308939925

Offset: 1

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn

Comments

See A192232.

Crossrefs

Cf. A192232, A192239.

Programs

Mathematica

q[x_] := x + 1;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := (x + n) p[n - 1, x] /; n > 1
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[
  Last[Most[
    FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
   20}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]  (* A192238 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]  (* A192239 *)
(* Peter J. C. Moses, Jun 25 2011 *)

Formula

Conjecture: a(n) +2*(-n+1)*a(n-1) +(n^2-3*n+1)*a(n-2)=0. - R. J. Mathar, May 04 2014

E.g.f.: 2 - (cosh((sqrt(5)/2)*log(1-x)) + (3/sqrt(5))*sinh((sqrt(5)/2)*log(1-x)))/(1-x)^(3/2). - Fabian Pereyra, Oct 28 2024