A192944 - OEIS

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A192944

Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x)=(x+1)(x+2)...(x+F(n+1)), where F=A000045, the Fibonacci sequence.

0, 1, 4, 19, 127, 1227, 17977, 407108, 14510651, 821907178, 74498246381, 10849935064552, 2545826568211757, 963950723522943935, 589590299737652176495, 582892188767255266969095, 931834379222684656945128850, 2409387593714287957763063565225

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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..97

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+4x

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

From these, read

A192943=(1,1,3,13,...) and A192944=(0,1,4,19,...)

MATHEMATICA

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

p[0, x]:= 1;

p[n_, x_]:= (x + Fibonacci[n+1])*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}] (* A192943 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192944 *)

CROSSREFS

Cf. A192232, A192744, A192943.

Sequence in context: A203236 A365961 A103468 * A306179 A330515 A330534

Adjacent sequences: A192941 A192942 A192943 * A192945 A192946 A192947

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved