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 A192462 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
 1, 1, 3, 18, 134, 1219, 13051, 160877, 2243285, 34910810, 599778960, 11274872675, 230192376755, 5072160696515, 119969157163845, 3031681775228370, 81517508176185730, 2323785190405594685, 70003126753631869325, 2222084456557049981875 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The polynomial p(n,x) is defined by recursively by p(n,x)=(nx+1)*p(n-1,x) with p[0,x]=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232. LINKS Table of n, a(n) for n=0..19. EXAMPLE The first four polynomials p(n,x) and their reductions are as follows: p(0,x) = 1. p(1,x)=1+x -> 1+x. p(2,x)=(1+x)(1+2x) -> 3+5x. p(3,x)=(1+x)(1+2x)(1+3x) -> 18+29x. p(4,x)=(1+x)(1+2x)(1+3x)(1+4x) -> 134+217x. From these, read A192462=(1,1,3,18,134,...) and A192463=(0,1,5,29,217,...) MATHEMATICA q[x_] := x + 1; p[0, x_] := 1; p[n_, x_] := (n*x + 1)*p[n - 1, x] /; n > 0 Table[Simplify[p[n, x]], {n, 1, 5}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 20}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}] (* A192462 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}] (* A192463 *) CROSSREFS Cf. A192232, A192463. Sequence in context: A236342 A060909 A074545 * A168072 A251733 A355103 Adjacent sequences: A192459 A192460 A192461 * A192463 A192464 A192465 KEYWORD nonn AUTHOR Clark Kimberling, Jul 01 2011 STATUS approved

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Last modified October 2 23:30 EDT 2023. Contains 365841 sequences. (Running on oeis4.)