A085707 Triangular array A065547 unsigned and transposed.

1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 17, 17, 0, 1, 10, 55, 155, 155, 0, 1, 15, 135, 736, 2073, 2073, 0, 1, 21, 280, 2492, 13573, 38227, 38227, 0, 1, 28, 518, 6818, 60605, 330058, 929569, 929569, 0, 1, 36, 882, 16086, 211419, 1879038, 10233219, 28820619

Offset: 0

Author

Philippe Deléham, Jul 19 2003

Keywords

• nonn
• tabl

Examples

			1;
1,  0;
1,  1,  0;
1,  3,  3,   0;
1,  6, 17,  17,   0;
1, 10, 55, 155, 155, 0;
...
		

References

• Louis Comtet, Analyse Combinatoire, PUF, 1970, Tome 2, pp. 98-99.

Links

• J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
• J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

Crossrefs

Row sums Sum_{k>=0} T(n, k) = A006846(n), values of Hammersley's polynomial p_n(1).

Sum_{k>=0} 2^k*T(n, k) = A005647(n), Salie numbers.

Sum_{k>=0} 3^k*T(n, k) = A094408(n).

Sum_{k>=0} 4^k*T(n, k) = A000364(n), Euler numbers.

Cf. A006846, A005647, A000364, A065547, A000795.

Cf. A006846 A005647 A000364 A065547 A000795 A084938.

Programs

Mathematica

h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First // Abs // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 02 2013 *)

Formula

Sum_{k >= 0} (-1/2)^k*T(n, k) = (1/2)^n.

Sum_{k >= 0} (-1/6)^k*T(n, k) = (4^(n+1)- 1)/3*(6^n).

Equals A000035 DELTA [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...], where DELTA is Deléham's operator defined in A084938.

T(n,n-1) = A110501(n), Genocchi numbers of first kind of even index. - Philippe Deléham, Feb 16 2007