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 A193048 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments. 2
 1, 0, 1, 2, 8, 25, 68, 163, 357, 730, 1417, 2642, 4774, 8417, 14556, 24793, 41729, 69582, 115187, 189614, 310786, 507715, 827356, 1345697, 2185703, 3546350, 5749603, 9316428, 15089782, 24433615, 39554862, 64024437, 103620219, 167691032 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,x)+n(4-5*n^2+n^4)/120, with p(0,x)=1.  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744. LINKS Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1). FORMULA a(n) = 6*a(n-1)-14*a(n-2)+15*a(n-3)-5*a(n-4)-4*a(n-5)+4*a(n-6)-a(n-7). G.f.: (x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1) / ((x-1)^5*(x^2+x-1)). - Colin Barker, May 12 2014 MATHEMATICA q = x^2; s = x + 1; z = 40; p[0, x] := 1; p[n_, x_] := x*p[n - 1, x] + n (-1 + n^2) (-4 + n^2)/120; 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}] (* A193048 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193049 *) PROG (PARI) Vec((x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1)/((x-1)^5*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014 CROSSREFS Cf. A192232, A192744, A192951, A193049, A193046. Sequence in context: A198181 A066455 A066374 * A301819 A119854 A176855 Adjacent sequences:  A193045 A193046 A193047 * A193049 A193050 A193051 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 15 2011 STATUS approved

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Last modified March 31 16:23 EDT 2020. Contains 333151 sequences. (Running on oeis4.)