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Constant term in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.
3

%I #13 Nov 19 2024 01:06:22

%S 1,0,1,6,37,256,1999,17490,169895,1816320,21205745,268547510,

%T 3667187645,53722014720,840455448415,13985762375970,246675543859855,

%U 4596826887347200,90249727067243425,1861971659969854950,40274219840308939925

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

%C See A192232.

%F 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 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 q[x_] := x + 1;

%t p[0, x_] := 1; p[1, x_] := x;

%t p[n_, x_] := (x + n) p[n - 1, x] /; n > 1

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[

%t Last[Most[

%t FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,

%t 20}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}] (* A192238 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}] (* A192239 *)

%t (* _Peter J. C. Moses_, Jun 25 2011 *)

%Y Cf. A192232, A192239.

%K nonn,changed

%O 1,4

%A _Clark Kimberling_, Jun 26 2011