A141720 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

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.

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, 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

Cf. A008292, A019538.