A168289 T(n,k) = 4*A046802(n,k) - 3*A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 48, 114, 48, 1, 1, 109, 494, 494, 109, 1, 1, 234, 1847, 3472, 1847, 234, 1, 1, 487, 6381, 20339, 20339, 6381, 487, 1, 1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1, 1, 2017, 67360, 524848, 1377826, 1377826, 524848

Offset

0,5

Links

Table of n, a(n) for n=0..51.

Formula

E.g.f: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 3*exp(t*(1 + x)).

Example

Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  19,    19,      1;
     1,  48,   114,     48,      1;
     1, 109,   494,    494,    109,      1;
     1, 234,  1847,   3472,   1847,    234,     1;
     1, 487,  6381,  20339,  20339,   6381,   487,   1;
     1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1;
      ... reformatted. - Franck Maminirina Ramaharo, Oct 21 2018

Mathematica

p[t_] = 4*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 3*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

Prog

(Maxima)
A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 3*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

Crossrefs

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168290, A168291, A168292, A168293.

Sequence in context: A157632 A328888 A176125 * A141690 A318408 A146957

Adjacent sequences: A168286 A168287 A168288 * A168290 A168291 A168292

Keyword

nonn,easy,tabl

Author

Roger L. Bagula, Nov 22 2009

Extensions

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

Definition corrected by Georg Fischer, Jan 28 2026

Status

approved