A157744 A recursion triangle sequence: A(n,k) = A(n-1,k-1)+e(n-1,k) where e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k - j)^n, {j, 0, k}].

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 1, 2, 13, 17, 5, 1, 1, 2, 28, 79, 43, 6, 1, 1, 2, 59, 330, 381, 100, 7, 1, 1, 2, 122, 1250, 2746, 1572, 220, 8, 1, 1, 2, 249, 4415, 16869, 18365, 5865, 467, 9, 1, 1, 2, 504, 14857, 92649, 173059, 106599, 20473, 969, 10, 1

Offset

0,5

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..65.

Formula

A(n,k) = A(n-1,k-1)+e(n-1,k) where e(n,k) = Sum[(-1)^j Binomial[n + 1, j](k - j)^n, {j, 0, k}].

Example

{1},
{1, 1},
{1, 2, 1},
{1, 2, 3, 1},
{1, 2, 6, 4, 1},
{1, 2, 13, 17, 5, 1},
{1, 2, 28, 79, 43, 6, 1},
{1, 2, 59, 330, 381, 100, 7, 1},
{1, 2, 122, 1250, 2746, 1572, 220, 8, 1},
{1, 2, 249, 4415, 16869, 18365, 5865, 467, 9, 1},
{1, 2, 504, 14857, 92649, 173059, 106599, 20473, 969, 10, 1}

Mathematica

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

Crossrefs

Sequence in context: A225641 A116855 A173265 * A335545 A334997 A030111

Adjacent sequences: A157741 A157742 A157743 * A157745 A157746 A157747

Keyword

nonn,tabl

Author

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

Status

approved