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Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.
3

%I #16 Mar 30 2020 04:20:03

%S 1,0,1,1,-2,2,2,9,-6,6,9,-28,12,-24,24,44,185,100,60,-120,120,265,

%T -846,-690,-120,360,-720,720,1854,7777,2478,5250,-840,2520,-5040,5040,

%U 14833,-47384,33656,-40656,1680,-6720,20160,-40320,40320,133496,559953,-347832,181944,359856,15120,-60480,181440,-362880,362880

%N Triangle read by rows: T(n,k) = logarithmic polynomial A_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 0,1,

%e 1,-2,2,

%e 2,9,-6,6,

%e 9,-28,12,-24,24,

%e 44,185,100,60,-120,120,

%e 265,-846,-690,-120,360,-720,720,

%e ...

%p A260324 := 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+1)/(r-j*n+1)!,j=1..(r+1)/n) ;

%p %*r! ;

%p end if;

%p end proc:

%p for r from 0 to 20 do

%p for n from 1 to r+1 do

%p printf("%a,",A260324(n,r)) ;

%p end do:

%p printf("\n") ;

%p end do: # _R. J. Mathar_, Jul 24 2015

%t 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 Table[T[n, k] /. x -> 1, {k, 0, 9}, {n, 1, k + 1}] // Flatten (* _Jean-François Alcover_, Mar 30 2020 *)

%Y Rows, column sums give A000166, A002747, A002748, A002749.

%Y Cf. A260322, A260323, A260325.

%K sign,tabl

%O 1,5

%A _N. J. A. Sloane_, Jul 23 2015