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A193044
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Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
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2
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1, 0, 2, 5, 13, 28, 56, 105, 189, 330, 564, 949, 1579, 2606, 4276, 6987, 11383, 18506, 30042, 48719, 78951, 127880, 207062, 335195, 542533, 878028, 1420886, 2299265, 3720529, 6020200, 9741164, 15761829, 25503489, 41265846, 66769896
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OFFSET
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0,3
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COMMENTS
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The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(-1+n^2)/6, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.
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LINKS
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Table of n, a(n) for n=0..34.
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FORMULA
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a(n)=4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
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MATHEMATICA
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q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n (n^2 - 1)/6;
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}]
(* A193044 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A193045 *)
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CROSSREFS
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Cf. A192232, A192744, A192951, A193045.
Sequence in context: A126656 A026522 A216378 * A122491 A002559 A049097
Adjacent sequences: A193041 A193042 A193043 * A193045 A193046 A193047
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, Jul 15 2011
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STATUS
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approved
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