cp's OEIS Frontend

A260325 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.

%I A260325 #22 Apr 26 2023 06:14:02
%S A260325 1,2,1,5,2,2,16,9,6,6,65,28,12,24,24,326,185,140,60,120,120,1957,846,
%T A260325 750,120,360,720,720,13700,7777,2562,5250,840,2520,5040,5040,109601,
%U A260325 47384,47096,40656,1680,6720,20160,40320,40320,986410,559953,378072,181944,365904,15120,60480,181440,362880,362880
%N A260325 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.
%H A260325 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 A260325 Triangle begins:
%e A260325      1;
%e A260325      2,   1;
%e A260325      5,   2,   2;
%e A260325     16,   9,   6,   6;
%e A260325     65,  28,  12,  24,  24;
%e A260325    326, 185, 140,  60, 120, 120;
%e A260325   1957, 846, 750, 120, 360, 720, 720;
%e A260325   ...
%p A260325 A260325 := proc(n,r)
%p A260325     if r = 0 then
%p A260325         1 ;
%p A260325     elif n > r+1 then
%p A260325         0 ;
%p A260325     else
%p A260325         add( 1/(r-j*n+1)!,j=1..(r+1)/n) ;
%p A260325         %*r! ;
%p A260325     end if;
%p A260325 end proc:
%p A260325 for r from 0 to 20 do
%p A260325     for n from 1 to r+1 do
%p A260325         printf("%a,",A260325(n,r)) ;
%p A260325     end do:
%p A260325     printf("\n") ;
%p A260325 end do: # _R. J. Mathar_, Jul 24 2015
%t A260325 T[n_, k_] := Which[n == 0, 1, k > n+1, 0, True, Sum[1/(n-j*k+1)!, {j, 1, (n+1)/k}]*n!];
%t A260325 Table[T[n, k], {n, 0, 9}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Apr 25 2023 *)
%Y A260325 Rows, column sums give A000522, A002747, A002750, A002751.
%Y A260325 Main diagonal gives A000142.
%Y A260325 Cf. A260322, A260323, A260324.
%K A260325 sign,tabl
%O A260325 1,2
%A A260325 _N. J. A. Sloane_, Jul 23 2015