A260322 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=1.

1, -1, 2, 2, -6, 6, 0, 24, -24, 24, 9, -80, 60, -120, 120, 35, 450, 240, 360, -720, 720, 230, -2142, -2310, -840, 2520, -5040, 5040, 1624, 17696, 9744, 21840, -6720, 20160, -40320, 40320, 13209, -112464, 91224, -184464, 15120, -60480, 181440, -362880, 362880

Offset: 1

Author

N. J. A. Sloane, Jul 23 2015

Keywords

• sign
• tabl

Examples

			Triangle begins:
    1;
   -1,     2;
    2,    -6,     6;
    0,    24,   -24,   24;
    9,   -80,    60, -120,  120;
   35,   450,   240,  360, -720,   720;
  230, -2142, -2310, -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 A002741, A002742, A002743, A002744.

Main diagonal gives A000142.

Cf. A260323-A260325.

Programs

Maple

A260322 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( (-1)^(r-j*n)/(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,",A260322(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

Mathematica

T[n_, k_] := Which[n == 0, 1, k > n+1, 0, True,
   Sum[(-1)^(n-j*k)/(n-j*k)!/j, {j, 1, n/k}]] n!;
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 30 2023 *)