A192941 -id:A192941 - cp's OEIS frontend

A192942 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).

0, 2, 10, 62, 448, 3688, 34056, 348568, 3916352, 47919520, 634256480, 9029234720, 137569217280, 2233574372480, 38497936301440, 702049663399040, 13504656880506880, 273280886412413440, 5803407252377602560, 129044887279907315200

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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

			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,...)

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

Cf. A192232, A192744, A192941, A192950.

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

(* 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 *)

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

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

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

A192950 a(n) = A192942(n)/2.

Original entry on oeis.org

0, 1, 5, 31, 224, 1844, 17028, 174284, 1958176, 23959760, 317128240, 4514617360, 68784608640, 1116787186240, 19248968150720, 351024831699520, 6752328440253440, 136640443206206720, 2901703626188801280, 64522443639953657600

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

See A192942.

			(See A192942.)

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A192942.

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)))/10: n in [0..20]]; // G. C. Greubel, Jul 25 2019

(* 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*Gamma[n+2+s]/Gamma[s+2] - Sin[Pi*(s+3)]*Gamma[s+1]*Gamma[n+2-s]/(Pi*(s-1)))/10], {n, 0, 20}]] (* G. C. Greubel, Jul 25 2019 *)

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)))/10 ) \\ G. C. Greubel, Jul 25 2019

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)))/10 for n in (0..20)] # G. C. Greubel, Jul 25 2019

a(n) = 1/10*sqrt(5)*Gamma(n+2+sqrt(5))/Gamma(sqrt(5)+2) - 1/10*sin(Pi*(sqrt(5)+3))*Gamma(sqrt(5)+1)*Gamma(n+2-sqrt(5))/(Pi*(sqrt(5)-1)). - Vaclav Kotesovec, Oct 26 2012

cp's OEIS Frontend

A192942 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).

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