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