A168287 T(n,k) = 2*A046802(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 60, 26, 1, 1, 57, 252, 252, 57, 1, 1, 120, 931, 1746, 931, 120, 1, 1, 247, 3201, 10187, 10187, 3201, 247, 1, 1, 502, 10534, 53542, 89788, 53542, 10534, 502, 1, 1, 1013, 33698, 262466, 688976, 688976, 262466, 33698, 1013

Offset

0,5

Links

Table of n, a(n) for n=0..53.

Formula

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

Example

Triangle begins:
     1;
     1,    1;
     1,    4,     1;
     1,   11,    11,     1;
     1,   26,    60,    26,     1;
     1,   57,   252,   252,    57,     1;
     1,  120,   931,  1746,   931,   120,     1;
     1,  247,  3201, 10187, 10187,  3201,   247,   1;
     1,  502, 10534, 53542, 89788, 53542, 10534, 502, 1;
     ... reformatted. - Franck Maminirina Ramaharo, Oct 21 2018

Mathematica

p[t_] = 2*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 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) := 2*A046802(n + 1, k + 1) - 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, A168288, A168289, A168290, A168291, A168292, A168293.

Sequence in context: A324916 A156534 A375858 * A221987 A285357 A174526

Adjacent sequences: A168284 A168285 A168286 * A168288 A168289 A168290

Keyword

nonn,tabl,easy

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