A320080 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - k*log(1 + x)).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 2, 0, 1, 4, 15, 28, 4, 0, 1, 5, 28, 114, 172, 14, 0, 1, 6, 45, 296, 1152, 1328, 38, 0, 1, 7, 66, 610, 4168, 14562, 12272, 216, 0, 1, 8, 91, 1092, 11020, 73376, 220842, 132480, 600, 0, 1, 9, 120, 1778, 24084, 248870, 1550048, 3907656, 1633344, 6240, 0

Offset: 0

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

• nonn
• tabl

Examples

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k - 1)*x^2/2! + 2*k*(3*k^2 - 3*k + 1)*x^3/3! + 2*k*(12*k^3 - 18*k^2 + 11*k - 3)*x^4/4! + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   1,     6,     15,     28,      45,  ...
  0,   2,    28,    114,    296,     610,  ...
  0,   4,   172,   1152,   4168,   11020,  ...
  0,  14,  1328,  14562,  73376,  248870,  ...
		

Crossrefs

Columns k=0..5 give A000007, A006252, A088501, A335531, A354147, A365604.

Main diagonal gives A317172.

Cf. A048594, A048994, A094416, A320079, A334369.

Programs

Mathematica

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

Formula

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

A(n,k) = Sum_{j=0..n} Stirling1(n,j)*j!*k^j.

A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (-1)^(j-1) * (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022