A292717 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - x)/(1 - x)^k.

0, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 11, 6, 0, 1, 7, 26, 50, 24, 0, 1, 9, 47, 154, 274, 120, 0, 1, 11, 74, 342, 1044, 1764, 720, 0, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 0, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 0, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 362880

Offset

0,9

Links

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

Formula

E.g.f. of column k: -log(1 - x)/(1 - x)^k.

Example

E.g.f. of column k: A_k(x) = x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 6*k + 2)*x^3/3! + 2*(2*k^3 + 9*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
   0,    0,     0,     0,     0,      0,  ...
   1,    1,     1,     1,     1,      1,  ...
   1,    3,     5,     7,     9,     11,  ...
   2,   11,    26,    47,    74,    107,  ...
   6,   50,   154,   342,   638,   1066,  ...
  24,  274,  1044,  2754,  5944,  11274,  ...

Mathematica

Table[Function[k, n! SeriesCoefficient[-Log[1 - x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..11 give A104150, A000254, A001705, A001711 (with offset 1), A001716 (with offset 1), A001721 (with offset 1), A051524, A051545, A051560, A051562, A051564, A203147.

Rows n=0..3 give A000004, A000012, A005408, A080663 (with offset 0).

Main diagonal gives A058806.

Sequence in context: A360866 A081576 A330785 * A365727 A054654 A355423

Adjacent sequences: A292714 A292715 A292716 * A292718 A292719 A292720

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Sep 21 2017

Status

approved