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A260324 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.

%I A260324 #16 Mar 30 2020 04:20:03
%S A260324 1,0,1,1,-2,2,2,9,-6,6,9,-28,12,-24,24,44,185,100,60,-120,120,265,
%T A260324 -846,-690,-120,360,-720,720,1854,7777,2478,5250,-840,2520,-5040,5040,
%U A260324 14833,-47384,33656,-40656,1680,-6720,20160,-40320,40320,133496,559953,-347832,181944,359856,15120,-60480,181440,-362880,362880
%N A260324 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.
%H A260324 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 A260324 Triangle begins:
%e A260324 1,
%e A260324 0,1,
%e A260324 1,-2,2,
%e A260324 2,9,-6,6,
%e A260324 9,-28,12,-24,24,
%e A260324 44,185,100,60,-120,120,
%e A260324 265,-846,-690,-120,360,-720,720,
%e A260324 ...
%p A260324 A260324 := proc(n,r)
%p A260324     if r = 0 then
%p A260324         1 ;
%p A260324     elif n > r+1 then
%p A260324         0 ;
%p A260324     else
%p A260324         add( (-1)^(r-j*n+1)/(r-j*n+1)!,j=1..(r+1)/n) ;
%p A260324         %*r! ;
%p A260324     end if;
%p A260324 end proc:
%p A260324 for r from 0 to 20 do
%p A260324     for n from 1 to r+1 do
%p A260324         printf("%a,",A260324(n,r)) ;
%p A260324     end do:
%p A260324     printf("\n") ;
%p A260324 end do: # _R. J. Mathar_, Jul 24 2015
%t A260324 T[n_, k_] := If[k == 0, 1, If[n > k + 1, 0, k! Sum[(-x)^(k - j n + 1)/(k - j n + 1)!, {j, 1, (k + 1)/n}]]];
%t A260324 Table[T[n, k] /. x -> 1, {k, 0, 9}, {n, 1, k + 1}] // Flatten (* _Jean-François Alcover_, Mar 30 2020 *)
%Y A260324 Rows, column sums give A000166, A002747, A002748, A002749.
%Y A260324 Cf. A260322, A260323, A260325.
%K A260324 sign,tabl
%O A260324 1,5
%A A260324 _N. J. A. Sloane_, Jul 23 2015