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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.
2
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved