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

%I A192239 #12 Dec 04 2016 19:46:25
%S A192239 0,1,3,13,71,463,3497,29975,287265,3042545,35284315,444617525,
%T A192239 6048575335,88347242335,1378930649745,22903345844335,403342641729665,
%U A192239 7506843094993825,147226845692229875,3034786640911840925,65592491119118514375
%N A192239 Coefficient of x in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.
%C A192239 See A192232.
%F A192239 From _Vaclav Kotesovec_, Oct 20 2012: (Start)
%F A192239 Recurrence (for n>3): a(n) = 2*(n-1)*a(n-1) - (n^2-3*n+1)*a(n-2).
%F A192239 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).
%F A192239 a(n) ~ n!*n^(sqrt(5)/2 - 1/2)*(3*sqrt(5) - 5)/(10*Gamma((1 + sqrt(5))/2)).
%F A192239 (End)
%t A192239 q[x_] := x + 1;
%t A192239 p[0, x_] := 1; p[1, x_] := x;
%t A192239 p[n_, x_] := (x + n) p[n - 1, x] /; n > 1
%t A192239 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192239    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192239 t = Table[
%t A192239   Last[Most[
%t A192239     FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192239    20}]
%t A192239 Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]  (* A192238 *)
%t A192239 Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]  (* A192239 *)
%t A192239 (* by _Peter J. C. Moses_, Jun 25 2011 *)
%t A192239 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 *)
%t A192239 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 *)
%Y A192239 Cf. A192232, A192238.
%K A192239 nonn
%O A192239 1,3
%A A192239 _Clark Kimberling_, Jun 26 2011
%E A192239 Minor edits by _Vaclav Kotesovec_, Mar 31 2014