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A143197
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Triangle read by rows: imaginary part of polylog expansion of Eulerian numbers: p(x,n) = (1 - I*x)^(n + 1)*PolyLog(-n, I*x)/x.
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1
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1, 1, 1, 0, 1, 0, -1, 1, 0, -11, 0, 1, 0, -66, 0, 1, 1, 0, -302, 0, 57, 0, 1, 0, -1191, 0, 1191, 0, -1, 1, 0, -4293, 0, 15619, 0, -247, 0, 1, 0, -14608, 0, 156190, 0, -14608, 0, 1, 1, 0, -47840, 0, 1310354, 0, -455192, 0, 1013, 0, 1, 0, -152637, 0, 9738114, 0, -9738114, 0, 152637, 0, -1
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OFFSET
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0,10
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LINKS
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FORMULA
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T(n, k) = [x^k]( Im( p(x, n) ) ), where p(x, n) = (1 - i*x)^(n + 1)*PolyLog(-n, i*x)/x, and T(0, 0) = 1.
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EXAMPLE
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Coefficients of p(x, n):
i;
i, -1;
i, -4, -i;
i, -11, -11*i, 1;
i, -26, -66*i, 26, i;
i, -57, -302*i, 302, 57*i, -1;
i, -120, -1191*i, 2416, 1191*i, -120, -i;
i, -247, -4293*i, 15619, 15619*i, -4293, -247*i, 1;
i, -502, -14608*i, 88234, 156190*i, -88234, -14608*i, 502, i;
Triangle of imaginary coefficients of p(x, n) begins as:
1;
1;
1, 0;
1, 0, -1;
1, 0, -11, 0;
1, 0, -66, 0, 1;
1, 0, -302, 0, 57, 0;
1, 0, -1191, 0, 1191, 0, -1;
1, 0, -4293, 0, 15619, 0, -247, 0;
1, 0, -14608, 0, 156190, 0, -14608, 0, 1;
1, 0, -47840, 0, 1310354, 0, -455192, 0, 1013, 0;
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MATHEMATICA
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p[x_, n_]:= (1-I*x)^(n+1)*PolyLog[-n, I*x]/x;
T[n_, k_]:= Coefficient[Series[p[x, n], {x, 0, 60}], x, k];
Join[{1}, Table[Im[T[n, k]], {n, 12}, {k, 0, n-1}]]//Flatten
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PROG
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(SageMath)
def p(n, x): return (1-i*x)^(n+1)*polylog(-n, i*x)/x
def A143197(n, k): return imag( ( p(n, x) ).series(x, n+1).list()[k] )
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CROSSREFS
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KEYWORD
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tabf,sign,less
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AUTHOR
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EXTENSIONS
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The entries here are clearly all wrong (compare the example lines). What are the real parts? - N. J. A. Sloane, Oct 25 2008
The entries were those of A143196; entries replaced with those in the example by Georg Fischer, Nov 03 2018
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STATUS
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approved
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