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, 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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Adjacent sequences: A143194 A143195 A143196 * A143198 A143199 A143200

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