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A351339 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is Sum_{j=0..n} k^j * j^(n-j).

%I A351339 #16 Feb 09 2022 09:04:00
%S A351339 1,1,0,1,1,0,1,2,2,0,1,3,6,4,0,1,4,12,18,9,0,1,5,20,48,58,23,0,1,6,30,
%T A351339 100,201,202,66,0,1,7,42,180,516,885,762,210,0,1,8,56,294,1105,2756,
%U A351339 4116,3114,733,0,1,9,72,448,2094,6955,15300,20298,13754,2781,0
%N A351339 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is Sum_{j=0..n} k^j * j^(n-j).
%F A351339 G.f. of column k: Sum_{j>=0} (k*x)^j/(1 - j*x).
%e A351339 Square array begins:
%e A351339   1,  1,   1,    1,     1,     1,      1, ...
%e A351339   0,  1,   2,    3,     4,     5,      6, ...
%e A351339   0,  2,   6,   12,    20,    30,     42, ...
%e A351339   0,  4,  18,   48,   100,   180,    294, ...
%e A351339   0,  9,  58,  201,   516,  1105,   2094, ...
%e A351339   0, 23, 202,  885,  2756,  6955,  15198, ...
%e A351339   0, 66, 762, 4116, 15300, 45030, 112686, ...
%t A351339 T[0, k_] := 1; T[n_, 0] = 0; T[n_, k_] := Sum[k^j * j^(n - j), {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 08 2022 *)
%o A351339 (PARI) T(n, k) = sum(j=0, n, k^j*j^(n-j));
%Y A351339 Columns k=0..3 give A000007, A026898(n-1), A351279, A351282.
%Y A351339 Main diagonal gives A351340.
%K A351339 nonn,tabl,easy
%O A351339 0,8
%A A351339 _Seiichi Manyama_, Feb 08 2022