%I #7 Feb 22 2019 02:08:07
%S 1,2,1,7,6,1,28,30,24,1,131,162,153,120,1,746,918,1050,922,720,1,5097,
%T 5826,7332,7578,6459,5040,1,40440,43158,53856,63420,61224,51678,40320,
%U 1,363127,372546,435279,547180,592245,552498,465109,362880,1,3629302
%N A recursion triangle sequence based on the Eulerian numbers: A(n,k)=n*A(n-1,k-1)+k*Eulerian(n-1,k).
%C Row sums are: {1, 3, 14, 83, 567, 4357, 37333, 354097, 3690865, 41988961,...}.
%C I use here a different definition of the Eulerian numbers sum and different initial conditions.
%C I can't get it to match the numbers in table 21.3 page 471 which aren't as far as I can tell in the OEIS.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 470, Equation (38).
%F A(n,k)=n*A(n-1,k-1)+k*Eulerian(n-1,k).
%e {1},
%e {2, 1},
%e {7, 6, 1},
%e {28, 30, 24, 1},
%e {131, 162, 153, 120, 1},
%e {746, 918, 1050, 922, 720, 1},
%e {5097, 5826, 7332, 7578, 6459, 5040, 1},
%e {40440, 43158, 53856, 63420, 61224, 51678, 40320, 1},
%e {363127, 372546, 435279, 547180, 592245, 552498, 465109, 362880, 1},
%e {3629302, 3660486, 3990162, 4977550, 5912970, 6010098, 5528494, 4651098, 3628800, 1}
%t Clear[e, A, n, k];
%t e[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
%t A[1, n_] := 1;
%t A[n_, n_] := 1;
%t A[n_, k_] := n*A[n - 1, k - 1] + k*e[n - 1, k];
%t Table[Table[A[n, k], {k, 1, n}], {n, 1, 10}];
%t Flatten[%]
%K nonn,tabl
%O 0,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 05 2009