A168423 Triangle read by rows: expansion of e.g.f. (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, 1003, 1

Offset

0,14

Comments

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

Links

Table of n, a(n) for n=0..65.

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

Triangle begins:
   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

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

Sequence in context: A091258 A351568 A174544 * A350071 A336844 A072101

Adjacent sequences: A168420 A168421 A168422 * A168424 A168425 A168426

Keyword

sign,tabl

Author

Roger L. Bagula, Nov 25 2009

Status

approved