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A368504 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.

%I A368504 #20 Jan 06 2024 11:29:45
%S A368504 1,0,1,0,1,1,0,1,3,1,0,1,6,6,1,0,1,11,21,10,1,0,1,20,60,58,15,1,0,1,
%T A368504 37,161,244,141,21,1,0,1,70,428,900,857,318,28,1,0,1,135,1149,3164,
%U A368504 4225,2787,685,36,1,0,1,264,3132,10990,18945,18196,8704,1434,45,1
%N A368504 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.
%H A368504 OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>.
%F A368504 G.f. of column k: x*A_k(x)/((1-k*x) * (1-x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.
%F A368504 T(0,k) = 0^k; T(n,k) = k*T(n-1,k) + n^k.
%e A368504 Square array begins:
%e A368504   1,  0,   0,    0,     0,      0,      0, ...
%e A368504   1,  1,   1,    1,     1,      1,      1, ...
%e A368504   1,  3,   6,   11,    20,     37,     70, ...
%e A368504   1,  6,  21,   60,   161,    428,   1149, ...
%e A368504   1, 10,  58,  244,   900,   3164,  10990, ...
%e A368504   1, 15, 141,  857,  4225,  18945,  81565, ...
%e A368504   1, 21, 318, 2787, 18196, 102501, 536046, ...
%o A368504 (PARI) T(n, k) = sum(j=0, n, k^(n-j)*j^k);
%Y A368504 Columns k=0..5 give A000012, A000217, A047520, A066999, A067534, A218376.
%Y A368504 Main diagonal gives A368505.
%Y A368504 Cf. A368486.
%K A368504 nonn,tabl
%O A368504 0,9
%A A368504 _Seiichi Manyama_, Dec 27 2023