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

1, 1, 6, 38, 276, 2276, 21032, 215336, 2419824, 29611120, 391950240, 5579965600, 85018056640, 1380373170880, 23792373137280, 433881469662080, 8346202841391360, 168894762064666880, 3586667489988830720, 79753496814542988800

Offset

0,3

Comments

For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Formula

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

Conjecture: a(n) +(-2*n-1)*a(n-1) +(n^2-5)*a(n-2)=0. - R. J. Mathar, May 08 2014

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(0,x) = 1
p(1,x) = 2x+1 -> 1+2x
p(2,x) = (2x+1)(2x+2) -> 6+10x
p(3,x) = (2x+1)(2x+2)(2x+3) -> 38+62x
From these, read
A192941=(1,2,6,38,...) and A192942=(0,2,10,62,...)

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 26;
p[0, x]:= 1;
p[n_, x_]:= (2*x + n)*p[n-1, x];
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192941 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192942 *)
u2/2 (* A192950 *)
(* Additional programs *)
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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A192232, A192744, A192942.

Sequence in context: A079949 A026940 A082427 * A263855 A354326 A221283

Adjacent sequences: A192938 A192939 A192940 * A192942 A192943 A192944

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved