A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

1, 3, 2, 8, 6, 6, 24, 24, 24, 24, 89, 80, 60, 120, 120, 415, 450, 480, 360, 720, 720, 2372, 2142, 2730, 840, 2520, 5040, 5040, 16072, 17696, 10416, 21840, 6720, 20160, 40320, 40320, 125673, 112464, 151704, 184464, 15120, 60480, 181440, 362880, 362880

Offset: 1

Author

N. J. A. Sloane, Jul 23 2015

Keywords

• sign
• tabl

Examples

			Triangle begins:
1,
3,2,
8,6,6,
24,24,24,24,
89,80,60,120,120,
415,450,480,360,720,720,
2372,2142,2730,840,2520,5040,5040,
...
		

Links

• J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Crossrefs

Rows, column sums give A002104, A002742, A002745, A002746.

Cf. A260322-A260325.

Programs

Maple

A260323 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( 1/(r-j*n)!/j,j=1..(r)/n) ;
        %*r! ;
    end if;
end proc:
for r from 1 to 20 do
    for n from 1 to r do
        printf("%a,",A260323(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

Mathematica

T[n_, k_] := If[n == 0, 1, If[k > n+1, 0, Sum[1/(n - j*k)!/j, {j, 1, n/k}]]]*n!;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar *)