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

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