A168423 Triangle read by rows: expansion of (1 - x)/(exp(t)*(1 - x*exp(t*(1 - x)))).

1, -1, 1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 7, 1, -1, 1, 1, 21, 21, 1, 1, -1, 1, 51, 161, 51, 1, -1, 1, 1, 113, 813, 813, 113, 1, 1, -1, 1, 239, 3361, 7631, 3361, 239, 1, -1, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 1, -1, 1, 1003, 42865, 320107, 607009, 320107, 42865

Offset: 0

Author

Roger L. Bagula, Nov 25 2009

Keywords

• sign
• tabl

Comments

This sequence was derived from the Eulerian number umbral calculus expansion and A046802 by taking the exp(t) term and inverting it.

What is interesting here is the '1,-1' terms that appear.

I had thought I would get "1,5,1" not "1,7,1" from this function.

An OEIS search came up with A046739 which has the same internal symmetric number structure.

Inverse binomial transform of Eulerian numbers A123125. [Paul Barry, May 10 2011]

Examples

			{1},
{-1, 1},
{1, -1, 1},
{-1, 1, 1, 1},
{1, -1, 1, 7, 1},
{-1, 1, 1, 21, 21, 1},
{1, -1, 1, 51, 161, 51, 1},
{-1, 1, 1, 113, 813, 813, 113, 1},
{1, -1, 1, 239, 3361, 7631, 3361, 239, 1},
{-1, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1},
{1, -1, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1}
		

Crossrefs

Cf. A046802, A046739, A000166 (row sums), A123125.

Programs

Mathematica

p[t_] = (1 - x)/(Exp[t]*(1 - x*Exp[t*(1 - x)]))
a = Table[ CoefficientList[FullSimplify[ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];
Flatten[a]

Formula

E.g.f. sum(T(n,k) t^n/n! x^k) = p(x,t) = (1 - x)/(exp(t)*(1 - x*exp(t*(1 - x))))

T(n,k)=sum{j=0..n, (-1)^(n-j)*C(n,j)*A123125(j,k)}. [Paul Barry, May 10 2011]