OFFSET
0,14
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]
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]
EXAMPLE
{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}
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]
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Nov 25 2009
STATUS
approved