A168288 T(n,k) = 3*A046802(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 87, 37, 1, 1, 83, 373, 373, 83, 1, 1, 177, 1389, 2609, 1389, 177, 1, 1, 367, 4791, 15263, 15263, 4791, 367, 1, 1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1, 1, 1515, 50529, 393657, 1033401, 1033401, 393657, 50529

Offset

0,5

Links

Table of n, a(n) for n=0..52.

Formula

E.g.f.: 3*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*exp(t*(1 + x)).

Example

Triangle begins:
     1;
     1,   1;
     1,   5,     1;
     1,  15,    15,     1;
     1,  37,    87,    37,      1;
     1,  83,   373,   373,     83,     1;
     1, 177,  1389,  2609,   1389,   177,     1;
     1, 367,  4791, 15263,  15263,  4791,   367,   1;
     1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1;
      ... reformatted. - Franck Maminirina Ramaharo, Oct 21 2018

Mathematica

p[t_] = 3*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 2*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) := 3*A046802(n + 1, k + 1) - 2*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, A168289, A168290, A168291, A168292, A168293.

Sequence in context: A109960 A196019 A056940 * A157523 A141691 A157147

Adjacent sequences: A168285 A168286 A168287 * A168289 A168290 A168291

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