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

%I A192238 #13 Nov 19 2024 01:06:22
%S A192238 1,0,1,6,37,256,1999,17490,169895,1816320,21205745,268547510,
%T A192238 3667187645,53722014720,840455448415,13985762375970,246675543859855,
%U A192238 4596826887347200,90249727067243425,1861971659969854950,40274219840308939925
%N A192238 Constant term in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.
%C A192238 See A192232.
%F A192238 Conjecture: a(n) +2*(-n+1)*a(n-1) +(n^2-3*n+1)*a(n-2)=0. - _R. J. Mathar_, May 04 2014
%F A192238 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
%t A192238 q[x_] := x + 1;
%t A192238 p[0, x_] := 1; p[1, x_] := x;
%t A192238 p[n_, x_] := (x + n) p[n - 1, x] /; n > 1
%t A192238 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192238    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192238 t = Table[
%t A192238   Last[Most[
%t A192238     FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192238    20}]
%t A192238 Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]  (* A192238 *)
%t A192238 Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]  (* A192239 *)
%t A192238 (* _Peter J. C. Moses_, Jun 25 2011 *)
%Y A192238 Cf. A192232, A192239.
%K A192238 nonn
%O A192238 1,4
%A A192238 _Clark Kimberling_, Jun 26 2011