A192313 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.
1, 2, 5, 13, 34, 91, 247, 680, 1893, 5319, 15056, 42867, 122605, 351898, 1012729, 2920521, 8435362, 24392655, 70599403, 204472264
Offset: 1
Keywords
Examples
The first five polynomials at A157751 and their reductions are as follows: p0(x)=1 -> 1 p1(x)=2+x -> 2+x p2(x)=4+2x+x^2 -> 5+3x p3(x)=8+4x+4x^2+x^3 -> 13+10x p4(x)=16+8x+12x^2+4x^3+x^4 -> 34+31x. From these, we read A192313=(1,2,5,13,34,...) and A192314=(0,1,3,19,31,...)
Programs
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Mathematica
q[x_] := x + 1; p[0, x_] := 1; p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /; n > 0 (* A157751 *) Table[Simplify[p[n, x]], {n, 0, 5}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 20}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}] (* A192313 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}] (* A192337 *)
Formula
Empirical G.f.: x*(x+1)*(x^2-3*x+1)/(x^4+6*x^3+x^2-4*x+1). [Colin Barker, Nov 13 2012]
Comments