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A193474
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Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2*j)^n * x^(n-k).
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2
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1, 1, 0, 2, 0, 0, 6, 0, 1, 0, 24, 0, 8, 0, 0, 120, 0, 60, 0, 1, 0, 720, 0, 480, 0, 32, 0, 0, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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P(n, i) = A000111(n+1), where i is the imaginary unit.
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
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EXAMPLE
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The sequence of polynomials P(n, x) begins:
[0] 1;
[1] 1;
[2] 2;
[3] 6 + x^2;
[4] 24 + 8*x^2;
[5] 120 + 60*x^2 + x^4;
[6] 720 + 480*x^2 + 32*x^4;
[7] 5040 + 4200*x^2 + 546*x^4 + x^6.
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MAPLE
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A193474_polynom := proc(n, x) local k, j;
add(add((-1)^j*2^(-k)*binomial(k, j)*(k-2*j)^n*x^(n-k), j=0..k), k=0..n) end: seq(seq(coeff(A193474_polynom(n, x), x, i), i=0..n), n=0..10);
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MATHEMATICA
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p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Peter Luschny, Aug 01 2011
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STATUS
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approved
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