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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^j * j^k.
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%I #25 Jan 06 2024 11:29:23

%S 1,0,1,0,1,1,0,2,3,1,0,3,18,6,1,0,4,75,90,10,1,0,5,260,804,346,15,1,0,

%T 6,805,5444,5988,1146,21,1,0,7,2310,31180,70980,36363,3450,28,1,0,8,

%U 6279,159774,671180,710980,193827,9722,36,1,0,9,16392,756420,5468190,10436805,6019396,943968,26106,45,1

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

%H OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>.

%F G.f. of column k: k*x*A_k(k*x)/((1-x) * (1-k*x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

%e Square array begins:

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

%e 1, 1, 2, 3, 4, 5, ...

%e 1, 3, 18, 75, 260, 805, ...

%e 1, 6, 90, 804, 5444, 31180, ...

%e 1, 10, 346, 5988, 70980, 671180, ...

%e 1, 15, 1146, 36363, 710980, 10436805, ...

%o (PARI) T(n, k) = sum(j=0, n, k^j*j^k);

%Y Columns k=0..3 give A000012, A000217, A036800, A343808.

%Y Main diagonal gives A303991.

%Y Cf. A173018, A303990, A368479.

%K nonn,tabl,easy

%O 0,8

%A _Seiichi Manyama_, Dec 26 2023