A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)*n!*[t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

1, 1, 1, 3, 2, 1, 14, 9, 3, 1, 88, 56, 18, 4, 1, 694, 440, 140, 30, 5, 1, 6578, 4164, 1320, 280, 45, 6, 1, 72792, 46046, 14574, 3080, 490, 63, 7, 1, 920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1, 13109088, 8288136, 2620512, 552552, 87444, 11088, 1176, 108, 9, 1

Offset: 0

Author

Peter Luschny, Sep 15 2017

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1]
[1,           1]
[3,           2,      1]
[14,          9,      3,     1]
[88,         56,     18,     4,    1]
[694,       440,    140,    30,    5,   1]
[6578,     4164,   1320,   280,   45,   6,  1]
[72792,   46046,  14574,  3080,  490,  63,  7, 1]
[920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1]
		

Crossrefs

Row sums: A291978.

Columns: A007840 (c=1), A052860 (c=2).

Diagonal: A045943 (d=3).

Cf. A291980.

Programs

Maple

T_row := proc(n) exp(x*t)/(1 + log(1+x)): series(%, x, n+1):
seq(coeff((-1)^(n-k)*n!*coeff(%,x,n),t,k), k=0..n) end:
seq(T_row(n), n=0..9);

Mathematica

T[n_, k_] := Binomial[n, n - k]*Sum[j!*Abs[StirlingS1[n - k, j]], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)

Formula

T(n, k) = binomial(n, n - k)*Sum_{j=0..n - k} j!*abs(Stirling1(n - k, j)). - Detlef Meya, May 12 2024