A192239 Coefficient of x in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.

0, 1, 3, 13, 71, 463, 3497, 29975, 287265, 3042545, 35284315, 444617525, 6048575335, 88347242335, 1378930649745, 22903345844335, 403342641729665, 7506843094993825, 147226845692229875, 3034786640911840925, 65592491119118514375

Offset: 1

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn

Comments

See A192232.

Crossrefs

Cf. A192232, A192238.

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 *)
(* by Peter J. C. Moses, Jun 25 2011 *)
Flatten[{0,RecurrenceTable[{a[n]==2*(n-1)*a[n-1]-(n^2-3*n+1)*a[n-2],a[2]==1,a[3]==3},a,{n,2,20}]}] (* or *)
Flatten[{0,FullSimplify[Rest[Rest[CoefficientList[Series[-1/10*(Sqrt[5]+3)*Sqrt[5]*(x-1)^(Sqrt[5]/2-1/2)/(-1)^((1/2)*Sqrt[5]-1/2)-(1/10)*(Sqrt[5]-3)*Sqrt[5]*(x-1)^(-Sqrt[5]/2-1/2)/(-1)^(-Sqrt[5]/2-1/2), {x, 0, 20}], x]* Range[0, 20]!]]]}] (* Vaclav Kotesovec, Oct 20 2012 *)

Formula

From Vaclav Kotesovec, Oct 20 2012: (Start)

Recurrence (for n>3): a(n) = 2*(n-1)*a(n-1) - (n^2-3*n+1)*a(n-2).

E.g.f.: (for n>1): -1/10*(sqrt(5) + 3)*sqrt(5)*(x-1)^(sqrt(5)/2 - 1/2)/(-1)^((1/2)*sqrt(5) - 1/2) - (1/10)*(sqrt(5) - 3)*sqrt(5)*(x-1)^(-sqrt(5)/2 - 1/2)/(-1)^(-sqrt(5)/2 - 1/2).

a(n) ~ n!*n^(sqrt(5)/2 - 1/2)*(3*sqrt(5) - 5)/(10*Gamma((1 + sqrt(5))/2)).

(End)

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014