A157743 A recursion triangle sequence based on the Eulerian numbers: A(n,k)=n*A(n-1,k-1)+k*Eulerian(n-1,k).

1, 2, 1, 7, 6, 1, 28, 30, 24, 1, 131, 162, 153, 120, 1, 746, 918, 1050, 922, 720, 1, 5097, 5826, 7332, 7578, 6459, 5040, 1, 40440, 43158, 53856, 63420, 61224, 51678, 40320, 1, 363127, 372546, 435279, 547180, 592245, 552498, 465109, 362880, 1, 3629302

Offset

0,2

Comments

Row sums are: {1, 3, 14, 83, 567, 4357, 37333, 354097, 3690865, 41988961,...}.

I use here a different definition of the Eulerian numbers sum and different initial conditions.

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.

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 470, Equation (38).

Links

Table of n, a(n) for n=0..45.

Formula

A(n,k)=n*A(n-1,k-1)+k*Eulerian(n-1,k).

Example

{1},
{2, 1},
{7, 6, 1},
{28, 30, 24, 1},
{131, 162, 153, 120, 1},
{746, 918, 1050, 922, 720, 1},
{5097, 5826, 7332, 7578, 6459, 5040, 1},
{40440, 43158, 53856, 63420, 61224, 51678, 40320, 1},
{363127, 372546, 435279, 547180, 592245, 552498, 465109, 362880, 1},
{3629302, 3660486, 3990162, 4977550, 5912970, 6010098, 5528494, 4651098, 3628800, 1}

Mathematica

Clear[e, A, n, k];
e[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
A[1, n_] := 1;
A[n_, n_] := 1;
A[n_, k_] := n*A[n - 1, k - 1] + k*e[n - 1, k];
Table[Table[A[n, k], {k, 1, n}], {n, 1, 10}];
Flatten[%]

Crossrefs

Sequence in context: A244238 A019668 A091700 * A135895 A039814 A178120

Adjacent sequences: A157740 A157741 A157742 * A157744 A157745 A157746

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Mar 05 2009

Status

approved