A192943 Constant term of 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.

1, 1, 3, 13, 84, 799, 11614, 261871, 9310722, 526600361, 47689339307, 6941763106589, 1628280738899789, 616407665133432210, 376972626454916592035, 372661572610740328515040, 595723423648119559905487975, 1540281161085963627452726056250

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, A192944.

Sequence in context: A130406 A225236 A152789 * A366657 A369550 A125500

Adjacent sequences: A192940 A192941 A192942 * A192944 A192945 A192946

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved