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Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.
4

%I #16 Jun 25 2023 04:42:54

%S 1,3,2,8,6,6,24,24,24,24,89,80,60,120,120,415,450,480,360,720,720,

%T 2372,2142,2730,840,2520,5040,5040,16072,17696,10416,21840,6720,20160,

%U 40320,40320,125673,112464,151704,184464,15120,60480,181440,362880,362880

%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 3,2,

%e 8,6,6,

%e 24,24,24,24,

%e 89,80,60,120,120,

%e 415,450,480,360,720,720,

%e 2372,2142,2730,840,2520,5040,5040,

%e ...

%p A260323 := 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)!/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,",A260323(n,r)) ;

%p end do:

%p printf("\n") ;

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

%t T[n_, k_] := If[n == 0, 1, If[k > n+1, 0, Sum[1/(n - j*k)!/j, {j, 1, n/k}]]]*n!;

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 25 2023, after _R. J. Mathar_ *)

%Y Rows, column sums give A002104, A002742, A002745, A002746.

%Y Cf. A260322-A260325.

%K sign,tabl

%O 1,2

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