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A279927
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Expansion of e.g.f. arctan(x)*exp(x).
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4
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0, 1, 2, 1, -4, 9, 110, -279, -4520, 17265, 322618, -1638031, -35226860, 223578809, 5463436134, -41639195623, -1142009233872, 10162622387809, 309463272791538, -3149754003442847, -105510576441518164, 1208991988527548137, 44200537412519181278, -563099647603189449783
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n/2} binomial(n+1,2*k+1)*(-1)^k*((n-2*k)/(n+1))*(2k)!.
a(n+3) - a(n+2) + (n+1)*(n+2)*a(n+1) - (n+1)*(n+2)*a(n) = 1.
a(n+4) - 2*a(n+3) + (n^2+5*n+7)*a(n+2) - 2*(n+2)^2*a(n+1) + (n+1)*(n+2)*a(n) = 0. (End)
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EXAMPLE
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atan(x)*exp(x) = x + 2*x^2/2! + x^3/3! - 4*x^4/4! + 9*x^5/5! + ...
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MATHEMATICA
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CoefficientList[Series[Exp[x] ArcTan[x], {x, 0, 12}], x] Range[0, 12]!
Table[Sum[Binomial[n+1, 2k+1] (-1)^k (n-2k)/(n+1) (2k)!, {k, 0, n/2}], {n, 0, 12}] (* Emanuele Munarini, Dec 16 2017 *)
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PROG
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(PARI) x='x+O('x^33); concat([0], Vec(serlaplace(atan(x)*exp(x) ) ) ) \\ Joerg Arndt, Jan 06 2017
(Maxima) makelist(sum((-1)^k*binomial(n+1, 2*k+1)*(n-2*k)/(n+1)*(2*k)!, k, 0, floor(n/2)), n, 0, 12); /* Emanuele Munarini, Dec 16 2017 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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