A392822 - OEIS

%I #23 Feb 10 2026 22:16:35

%S 1,1,1,1,0,2,1,0,2,6,1,0,0,3,24,1,0,0,6,32,120,1,0,0,0,24,150,720,1,0,

%T 0,0,24,110,1524,5040,1,0,0,0,0,180,1320,12600,40320,1,0,0,0,0,120,

%U 1260,13916,147328,362880,1,0,0,0,0,0,1440,9450,142464,1705536,3628800

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. 1/(1 - x*(-log(1-x))^k).

%H Vincenzo Librandi, <a href="/A392822/b392822.txt">Table of n, a(n) for n = 0..495</a> (rows 0..31, flattened)

%F A(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (k*j)! * |Stirling1(n-j,k*j)|/(n-j)!.

%e Square array begins:

%e     1,    1,    1,    1,    1,   1, 1, ...

%e     1,    0,    0,    0,    0,   0, 0, ...

%e     2,    2,    0,    0,    0,   0, 0, ...

%e     6,    3,    6,    0,    0,   0, 0, ...

%e    24,   32,   24,   24,    0,   0, 0, ...

%e   120,  150,  110,  180,  120,   0, 0, ...

%e   720, 1524, 1320, 1260, 1440, 720, 0, ...

%t a[n_,k_]:=n! Sum[(k j)! Abs[StirlingS1[n-j,k j]]/(n-j)!,{j,0,Floor[n/(k+1)]}];

%t seq78=Module[{N=10,K=10,A},A=Table[a[n,k],{n,0,N},{k,0,K}];Flatten[Table[A[[i+1,s-i+1]],{s,0,N+K},{i,Max[0,s-K],Min[s,N]}]][[1;;78]]]; seq78 (* _Vincenzo Librandi_, Feb 03 2026 *)

%o (PARI) a(n, k) = n!*sum(j=0, n\(k+1), (k*j)!*abs(stirling(n-j, k*j, 1))/(n-j)!);

%Y Columns k=0..3 give A000142, A052830, A392823, A392824.

%K nonn,tabl,easy

%O 0,6

%A _Seiichi Manyama_, Jan 24 2026