A141720 Triangle of coefficients of the polynomials (1 - x)^n*A(n,x/(1 - x)), where A(n,x) are the Eulerian polynomials of A008292.

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

Offset: 1

Author

Roger L. Bagula, Sep 11 2008

Keywords

• sign
• tabf

Comments

Row sums are one.

Row n gives the coefficients in the expansion of Sum_{j=1..n} A008292(n,j)*x^j*(1 - x)^(n - j).

The coefficients of the polynomials (1 + x)^n*A(n,x/(1 + x)) are listed in A019538.

Examples

			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;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
• Eric Weisstein's World of Mathematics, Polylogarithm

Crossrefs

Cf. A008292, A019538, A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A144387, A144400, A174128.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
f:= func< n,x | n eq 0 select 1 else (&+[EulerianNumber(n,j-1)*x^j*(1-x)^(n-j): j in [1..n]]) >;
A141720:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A141720(n,k): k in [0..2*Floor((n+1)/2)-1], n in [1..15]]; // G. C. Greubel, Dec 30 2024

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

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 A141720(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([A141720(n) for n in range(1,13)]) # G. C. Greubel, Jul 15 2021

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

Extensions

Edited by Peter Bala, Jul 04 2012

Edited, and extra term removed by Franck Maminirina Ramaharo, Oct 24 2018