A253668 Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.

0, 0, 1, 0, 1, -1, 0, 1, 1, 2, 0, 1, 3, -1, -6, 0, 1, 5, 2, 2, 24, 0, 1, 7, 11, -2, -6, -120, 0, 1, 9, 26, 6, 4, 24, 720, 0, 1, 11, 47, 50, -6, -12, -120, -5040, 0, 1, 13, 74, 154, 24, 12, 48, 720, 40320, 0, 1, 15, 107, 342, 274, -24, -36, -240, -5040, -362880

Offset

0,10

Links

Table of n, a(n) for n=0..65.

Formula

T(n,n) = A000254(n).

Example

Square array starts:
[n\k][0  1   2   3    4     5     6]
[0]   0, 1, -1,  2,  -6,   24, -120, ...
[1]   0, 1,  1, -1,   2,   -6,   24, ...
[2]   0, 1,  3,  2,  -2,    4,  -12, ...
[3]   0, 1,  5, 11,   6,   -6,   12, ...
[4]   0, 1,  7, 26,  50,   24,  -24, ...
[5]   0, 1,  9, 47, 154,  274,  120, ...
[6]   0, 1, 11, 74, 342, 1044, 1764, ...
The first few rows as a triangle:
0,
0, 1,
0, 1, -1,
0, 1,  1,  2,
0, 1,  3, -1, -6,
0, 1,  5,  2,  2, 24,
0, 1,  7, 11, -2, -6, -120,
0, 1,  9, 26,  6,  4,   24, 720.

Maple

T := (n, k) -> k!*coeff(series(ln(x+1)*add(binomial(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;

Crossrefs

Cf. A000254.

Sequence in context: A239498 A079219 A197707 * A216220 A334892 A216235

Adjacent sequences: A253665 A253666 A253667 * A253669 A253670 A253671

Keyword

sign,tabl

Author

Peter Luschny, Jan 18 2015

Status

approved