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A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.

%I A343863 #23 May 03 2021 08:20:51
%S A343863 1,1,2,1,2,3,1,2,5,4,1,2,9,16,5,1,2,17,82,65,6,1,2,33,460,1313,326,7,
%T A343863 1,2,65,2674,29441,32826,1957,8,1,2,129,15796,684545,3680126,1181737,
%U A343863 13700,9,1,2,257,94042,16175105,427840626,794907217,57905114,109601,10
%N A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.
%H A343863 Seiichi Manyama, <a href="/A343863/b343863.txt">Antidiagonals n = 0..59, flattened</a>
%F A343863 T(0,k) = 1 and T(n,k) = n^k * T(n-1,k) + 1 for n > 0.
%e A343863 Square array begins:
%e A343863   1,   1,     1,       1,         1,           1, ...
%e A343863   2,   2,     2,       2,         2,           2, ...
%e A343863   3,   5,     9,      17,        33,          65, ...
%e A343863   4,  16,    82,     460,      2674,       15796, ...
%e A343863   5,  65,  1313,   29441,    684545,    16175105, ...
%e A343863   6, 326, 32826, 3680126, 427840626, 50547203126, ...
%t A343863 T[n_, k_] := Sum[(n!/j!)^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 03 2021 *)
%o A343863 (PARI) T(n, k) = sum(j=0, n, (n!/j!)^k);
%Y A343863 Columns 0..3 give A000027(n+1), A000522, A006040, A217284.
%Y A343863 Main diagonal gives A336247.
%Y A343863 Cf. A291556.
%K A343863 nonn,tabl
%O A343863 0,3
%A A343863 _Seiichi Manyama_, May 02 2021