A260325 - OEIS

%I #22 Apr 26 2023 06:14:02

%S 1,2,1,5,2,2,16,9,6,6,65,28,12,24,24,326,185,140,60,120,120,1957,846,

%T 750,120,360,720,720,13700,7777,2562,5250,840,2520,5040,5040,109601,

%U 47384,47096,40656,1680,6720,20160,40320,40320,986410,559953,378072,181944,365904,15120,60480,181440,362880,362880

%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      2,   1;

%e      5,   2,   2;

%e     16,   9,   6,   6;

%e     65,  28,  12,  24,  24;

%e    326, 185, 140,  60, 120, 120;

%e   1957, 846, 750, 120, 360, 720, 720;

%e   ...

%p A260325 := 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)!,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,",A260325(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, Sum[1/(n-j*k+1)!, {j, 1, (n+1)/k}]*n!];

%t Table[T[n, k], {n, 0, 9}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Apr 25 2023 *)

%Y Rows, column sums give A000522, A002747, A002750, A002751.

%Y Main diagonal gives A000142.

%Y Cf. A260322, A260323, A260324.

%K sign,tabl

%O 1,2

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