A253669 Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)), n>=0, k>=0.

0, 0, 1, 0, 1, -1, 0, 1, 3, 2, 0, 1, 7, -4, -6, 0, 1, 11, 26, 10, 24, 0, 1, 15, 74, -46, -36, -120, 0, 1, 19, 146, 342, 144, 168, 720, 0, 1, 23, 242, 1066, -756, -624, -960, -5040, 0, 1, 27, 362, 2414, 5944, 2844, 3408, 6480, 40320, 0, 1, 31, 506, 4578, 19524

Offset: 0

Author

Peter Luschny, Jan 18 2015

Keywords

• sign
• tabl

Examples

			Square array starts:
[n\k][0   1   2    3     4      5       6]
[0]   0,  1, -1,   2,   -6,    24,   -120, ...
[1]   0,  1,  3,  -4,   10,   -36,    168, ...
[2]   0,  1,  7,  26,  -46,   144,   -624, ...
[3]   0,  1, 11,  74,  342,  -756,   2844, ...
[4]   0,  1, 15, 146, 1066,  5944, -15768, ...
[5]   0,  1, 19, 242, 2414, 19524, 127860, ...
[6]   0,  1, 23, 362, 4578, 48504, 434568, ...
The first few rows as a triangle:
0,
0, 1,
0, 1, -1,
0, 1,  3,   2,
0, 1,  7,  -4,  -6,
0, 1, 11,  26,  10,  24,
0, 1, 15,  74, -46, -36, -120,
0, 1, 19, 146, 342, 144,  168, 720.
		

Crossrefs

Cf. A098118.

Programs

Maple

T := (n,k) -> k!*coeff(series(ln(x+1)*add(binomial(2*n,j)*x^j, j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k), k=0..6)) od;

Formula

T(n,n) = A098118(n).