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A192239
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Coefficient of x in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.
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4
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0, 1, 3, 13, 71, 463, 3497, 29975, 287265, 3042545, 35284315, 444617525, 6048575335, 88347242335, 1378930649745, 22903345844335, 403342641729665, 7506843094993825, 147226845692229875, 3034786640911840925, 65592491119118514375
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Recurrence (for n>3): a(n) = 2*(n-1)*a(n-1) - (n^2-3*n+1)*a(n-2).
E.g.f.: (for n>1): -1/10*(sqrt(5) + 3)*sqrt(5)*(x-1)^(sqrt(5)/2 - 1/2)/(-1)^((1/2)*sqrt(5) - 1/2) - (1/10)*(sqrt(5) - 3)*sqrt(5)*(x-1)^(-sqrt(5)/2 - 1/2)/(-1)^(-sqrt(5)/2 - 1/2).
a(n) ~ n!*n^(sqrt(5)/2 - 1/2)*(3*sqrt(5) - 5)/(10*Gamma((1 + sqrt(5))/2)).
(End)
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MATHEMATICA
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q[x_] := x + 1;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := (x + n) p[n - 1, x] /; n > 1
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}] (* A192238 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}] (* A192239 *)
Flatten[{0, RecurrenceTable[{a[n]==2*(n-1)*a[n-1]-(n^2-3*n+1)*a[n-2], a[2]==1, a[3]==3}, a, {n, 2, 20}]}] (* or *)
Flatten[{0, FullSimplify[Rest[Rest[CoefficientList[Series[-1/10*(Sqrt[5]+3)*Sqrt[5]*(x-1)^(Sqrt[5]/2-1/2)/(-1)^((1/2)*Sqrt[5]-1/2)-(1/10)*(Sqrt[5]-3)*Sqrt[5]*(x-1)^(-Sqrt[5]/2-1/2)/(-1)^(-Sqrt[5]/2-1/2), {x, 0, 20}], x]* Range[0, 20]!]]]}] (* Vaclav Kotesovec, Oct 20 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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