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A192936 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) = Prod_{k=1..n} (x+k). 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

G. C. Greubel, Table of n, a(n) for n = 0..100

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) -> 1 + x

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

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

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: A233824 A003319 A192239 * A000261 A111140 A302699

Adjacent sequences:  A192933 A192934 A192935 * A192937 A192938 A192939

KEYWORD

nonn,changed

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)