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A192936 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=(x+1)(x+2)...(x+n). 1
1, 1, 3, 13, 71, 463, 3497, 29975, 287265, 3042545, 35284315, 444617525, 6048575335, 88347242335, 1378930649745, 22903345844335, 403342641729665, 7506843094993825, 147226845692229875, 3034786640911840925, 65592491119118514375 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

Essentially the same as A192239.- R. J. Mathar, Aug 10 2011

LINKS

Table of n, a(n) for n=0..20.

FORMULA

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

EXAMPLE

The first four polynomials p(n,x) and their reductions are as follows:

p(0,x)=1

p(1,x)=x+1 -> x+1

p(2,x)=(x+1)(x+2) -> 3+4x

p(3,x)=(x+1)(x+2)(x+3) -> 13+19x

From these, read

A192936=(1,1,3,13,...) and A192239=(0,1,3,13,...)

MATHEMATICA

q = x^2; s = x + 1; z = 26;

p[0, x] := 1;

p[n_, x_] := (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}];

Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

  (* A192936 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

  (* A192239 *)

CROSSREFS

Cf. A192232, A192744, A192239.

Sequence in context: A158882 A233824 A192239 * A000261 A111140 A137983

Adjacent sequences:  A192933 A192934 A192935 * A192937 A192938 A192939

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified April 23 04:33 EDT 2014. Contains 240909 sequences.