A168291 T(n,k) = 4*A046802(n,k) - 2*A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 32, 82, 32, 1, 1, 65, 330, 330, 65, 1, 1, 130, 1159, 2304, 1159, 130, 1, 1, 259, 3801, 13195, 13195, 3801, 259, 1, 1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1, 1, 1029, 37212, 319332, 889230, 889230, 319332, 37212

Offset

0,5

Links

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

Formula

E.g.f.: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - exp(t*(1 + x)).

Example

Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  15,    15,     1;
     1,  32,    82,    32,      1;
     1,  65,   330,   330,     65,     1;
     1, 130,  1159,  2304,   1159,   130,     1;
     1, 259,  3801, 13195,  13195,  3801,   259,   1;
     1, 516, 12016, 67316, 117170, 67316, 12016, 516, 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) := 4*A046802(n + 1, k + 1) - 2*A008518(n, k) - 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, A168292, A168293.

Sequence in context: A152238 A295985 A086645 * A154980 A166344 A146766

Adjacent sequences: A168288 A168289 A168290 * A168292 A168293 A168294

Keyword

nonn,easy,less,tabl

Author

Roger L. Bagula, Nov 22 2009

Extensions

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

Definition corrected by Georg Fischer, Jan 28 2026

Status

approved