login
A143197
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.
1
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
OFFSET
0,10
FORMULA
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.
EXAMPLE
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;
MATHEMATICA
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
PROG
(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] )
flatten([1]+[[A143197(n, k) for k in range(n)] for n in (1..12)]) # G. C. Greubel, Jun 09 2024
CROSSREFS
Cf. A060187.
Sequence in context: A179920 A216726 A323169 * A138066 A173189 A115595
KEYWORD
tabf,sign,less
AUTHOR
Roger L. Bagula, Oct 19 2008
EXTENSIONS
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
Edited by G. C. Greubel, Jun 09 2024
STATUS
approved