A168293 T(n,k) = 12*A046802(n,k) - 9*A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 14, 1, 1, 33, 33, 1, 1, 64, 186, 64, 1, 1, 119, 724, 724, 119, 1, 1, 222, 2415, 5120, 2415, 222, 1, 1, 421, 7491, 28799, 28799, 7491, 421, 1, 1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1, 1, 1587, 66342, 649554, 1934544, 1934544, 649554

Offset

0,5

Links

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

Formula

E.g.f.: 12*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 9*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - 2*exp(t*(1 + x)).

Example

Triangle begins:
    1;
    1,   1;
    1,  14,     1;
    1,  33,    33,      1;
    1,  64,   186,     64,      1;
    1, 119,   724,    724,    119,      1;
    1, 222,  2415,   5120,   2415,    222,     1;
    1, 421,  7491,  28799,  28799,   7491,   421,   1;
    1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1:
     ... reformatted. - Franck Maminirina Ramaharo, Oct 21 2018

Prog

(Maxima)
A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
T(n, k) := 12*A046802(n + 1, k + 1) - 9*A008518(n, k) - 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, A168288, A168289, A168290, A168291, A168292.

Sequence in context: A040197 A040196 A168622 * A157633 A157278 A144441

Adjacent sequences: A168290 A168291 A168292 * A168294 A168295 A168296

Keyword

nonn,tabl,easy,less

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