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

%I A260322 #16 Apr 30 2023 11:17:14
%S A260322 1,-1,2,2,-6,6,0,24,-24,24,9,-80,60,-120,120,35,450,240,360,-720,720,
%T A260322 230,-2142,-2310,-840,2520,-5040,5040,1624,17696,9744,21840,-6720,
%U A260322 20160,-40320,40320,13209,-112464,91224,-184464,15120,-60480,181440,-362880,362880
%N A260322 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=1.
%H A260322 J. M. Gandhi, <a href="/A002741/a002741.pdf">On logarithmic numbers</a>, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]
%e A260322 Triangle begins:
%e A260322     1;
%e A260322    -1,     2;
%e A260322     2,    -6,     6;
%e A260322     0,    24,   -24,   24;
%e A260322     9,   -80,    60, -120,  120;
%e A260322    35,   450,   240,  360, -720,   720;
%e A260322   230, -2142, -2310, -840, 2520, -5040, 5040;
%e A260322   ...
%p A260322 A260322 := proc(n,r)
%p A260322     if r = 0 then
%p A260322         1 ;
%p A260322     elif n > r+1 then
%p A260322         0 ;
%p A260322     else
%p A260322         add( (-1)^(r-j*n)/(r-j*n)!/j,j=1..(r)/n) ;
%p A260322         %*r! ;
%p A260322     end if;
%p A260322 end proc:
%p A260322 for r from 1 to 20 do
%p A260322     for n from 1 to r do
%p A260322         printf("%a,",A260322(n,r)) ;
%p A260322     end do:
%p A260322     printf("\n") ;
%p A260322 end do: # _R. J. Mathar_, Jul 24 2015
%t A260322 T[n_, k_] := Which[n == 0, 1, k > n+1, 0, True,
%t A260322    Sum[(-1)^(n-j*k)/(n-j*k)!/j, {j, 1, n/k}]] n!;
%t A260322 Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2023 *)
%Y A260322 Rows, column sums give A002741, A002742, A002743, A002744.
%Y A260322 Main diagonal gives A000142.
%Y A260322 Cf. A260323-A260325.
%K A260322 sign,tabl
%O A260322 1,3
%A A260322 _N. J. A. Sloane_, Jul 23 2015