OFFSET
1,7
COMMENTS
LINKS
G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
Eric Weisstein's World of Mathematics, Polylogarithm
FORMULA
Row n is generated by the polynomial (1 - 2*x)^(n+1)*Li(-n, x/(1-x))/(1 - x), where Li(n, z) is the polylogarithm function.
Also generated by Sum_{k=0..n} (eulerian(n,k)*Sum_{l=0..n} (-1)^l*(n - l + 1)*(2 - x)^l*C(l + 1, k)). - Mourad Rahmani (mrahmani(AT)usthb.dz), Jul 22 2010
E.g.f.: (x*exp(2*x*y) - x*exp(y))/(x*exp(y) - (1 - x)*exp(2*x*y)). - Franck Maminirina Ramaharo, Oct 24 2018
EXAMPLE
Triangle begins:
0, 1;
0, 1;
0, 1, 2, -2;
0, 1, 8, -8;
0, 1, 22, -6, -32, 16;
0, 1, 52, 84, -272, 136;
0, 1, 114, 606, -1168, -96, 816, -272;
0, 1, 240, 2832, -2176, -8832, 11904, -3968;
0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936;
0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896;
...
MAPLE
CL := p -> PolynomialTools:-CoefficientList(p, x): flatten := seq -> ListTools:-Flatten(seq): flatten([seq(CL(add(A008292(n, j)*x^j*(1-x)^(n-j), j=1..n)), n=1..10)]); # Peter Luschny, Oct 25 2018
MATHEMATICA
Table[CoefficientList[FullSimplify[(1-2x)^(1+n)*PolyLog[-n, x/(1-x)]/(1-x)], x], {n, 1, 10}]//Flatten
PROG
(Sage)
def A(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k))
def p(n, x): return sum( A(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (1..12)]) # G. C. Greubel, Jul 15 2021
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Roger L. Bagula, Sep 11 2008
EXTENSIONS
Edited by Peter Bala, Jul 04 2012
Edited, and extra term removed by Franck Maminirina Ramaharo, Oct 24 2018
STATUS
approved