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A192942
Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).
3
0, 2, 10, 62, 448, 3688, 34056, 348568, 3916352, 47919520, 634256480, 9029234720, 137569217280, 2233574372480, 38497936301440, 702049663399040, 13504656880506880, 273280886412413440, 5803407252377602560, 129044887279907315200
OFFSET
0,2
COMMENTS
For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
LINKS
FORMULA
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_]:= (2x+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 *)
(* Second program *)
With[{s = Sqrt[5]}, Table[FullSimplify[(s*Gamma[n+2+s]/Gamma[s+2] - Sin[Pi*(s+3)]*Gamma[s+1]*Gamma[n+2-s]/(Pi*(s-1)))/5], {n, 0, 20}]] (* G. C. Greubel, Jul 26 2019 *)
PROG
(PARI) default(realprecision, 100); vector(20, n, n--; s=sqrt(5); round(s*gamma(n+2+s)/gamma(s+2) - sin(Pi*(s+3))*gamma(s+1)*gamma(n+2-s)/(Pi*(s-1)))/5 ) \\ G. C. Greubel, Jul 25 2019
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); s:=Sqrt(5); [Round(s*Gamma(n+2+s)/Gamma(s+2) - Sin(Pi(R)*(s+3))*Gamma(s+1) *Gamma(n+2-s)/(Pi(R)*(s-1)))/5: n in [0..20]]; // G. C. Greubel, Jul 25 2019
(Sage) s=sqrt(5); [round(s*gamma(n+2+s)/gamma(s+2) - sin(pi*(s+3))* gamma(s+1)*gamma(n+2-s)/(pi*(s-1)))/5 for n in (0..20)] # G. C. Greubel, Jul 25 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved