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A192941 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).

%I A192941 #15 Sep 08 2022 08:45:58
%S A192941 1,1,6,38,276,2276,21032,215336,2419824,29611120,391950240,5579965600,
%T A192941 85018056640,1380373170880,23792373137280,433881469662080,
%U A192941 8346202841391360,168894762064666880,3586667489988830720,79753496814542988800
%N A192941 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).
%C A192941 For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
%H A192941 G. C. Greubel, <a href="/A192941/b192941.txt">Table of n, a(n) for n = 0..445</a>
%F A192941 a(n) = 1/10*(5-sqrt(5))*Gamma(n+2+sqrt(5))/Gamma(2+sqrt(5)) + 1/10*sin(Pi*(3+sqrt(5)))*(1+sqrt(5))*Gamma(1+sqrt(5))*Gamma(n+2-sqrt(5))/(Pi*(sqrt(5)-1)). - _Vaclav Kotesovec_, Oct 26 2012
%F A192941 Conjecture: a(n) +(-2*n-1)*a(n-1) +(n^2-5)*a(n-2)=0. - _R. J. Mathar_, May 08 2014
%e A192941 The first four polynomials p(n,x) and their reductions are as follows:
%e A192941 p(0,x) = 1
%e A192941 p(1,x) = 2x+1 -> 1+2x
%e A192941 p(2,x) = (2x+1)(2x+2) -> 6+10x
%e A192941 p(3,x) = (2x+1)(2x+2)(2x+3) -> 38+62x
%e A192941 From these, read
%e A192941 A192941=(1,2,6,38,...) and A192942=(0,2,10,62,...)
%t A192941 (* First program *)
%t A192941 q = x^2; s = x + 1; z = 26;
%t A192941 p[0, x]:= 1;
%t A192941 p[n_, x_]:= (2*x + n)*p[n-1, x];
%t A192941 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192941 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192941 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192941 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192941 *)
%t A192941 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192942 *)
%t A192941 u2/2 (* A192950 *)
%t A192941 (* Additional programs *)
%t A192941 With[{s = Sqrt[5]}, Table[FullSimplify[(s*(s-1)*Gamma[n+2+s]/Gamma[s+2] + Sin[Pi*(s+3)]*Gamma[s+2]*Gamma[n+2-s]/(Pi*(s-1)))/10], {n, 0, 20}]] (* _G. C. Greubel_, Jul 25 2019 *)
%o A192941 (PARI) default(realprecision, 100); vector(20, n, n--; s=sqrt(5); round((s^2-s)*gamma(n+2+s)/gamma(s+2) + sin(Pi*(s+3))*gamma(s+2)* gamma(n+2-s)/(Pi*(s-1)))/10 ) \\ _G. C. Greubel_, Jul 25 2019
%o A192941 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); s:=Sqrt(5); [Round(s*(s-1)*Gamma(n+2+s)/Gamma(s+2) + Sin(Pi(R)*(s+3))*Gamma(s+2) *Gamma(n+2-s)/(Pi(R)*(s-1)))/10: n in [0..20]]; // _G. C. Greubel_, Jul 25 2019
%o A192941 (Sage) s=sqrt(5); [round(s*(s-1)*gamma(n+2+s)/gamma(s+2) + sin(pi*(s+3))* gamma(s+2)*gamma(n+2-s)/(pi*(s-1)))/10 for n in (0..20)] # _G. C. Greubel_, Jul 25 2019
%Y A192941 Cf. A192232, A192744, A192942.
%K A192941 nonn
%O A192941 0,3
%A A192941 _Clark Kimberling_, Jul 13 2011