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%I #15 Dec 18 2023 13:22:53
%S 1,1,1,1,1,1,1,1,2,1,1,1,3,6,1,1,1,4,13,24,1,1,1,5,22,75,120,1,1,1,6,
%T 33,160,541,720,1,1,1,7,46,285,1456,4683,5040,1,1,1,8,61,456,3081,
%U 15904,47293,40320,1,1,1,9,78,679,5656,40005,202672,545835,362880,1
%N A(n, k) = A_{n}(k) where A_{n}(x) are the Eulerian polynomials, square array read by ascending antidiagonals, for n >= 0 and k >= 0.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>.
%F A(n, k) = Sum_{j=0..k} a(k, j)*n^j where a(k, j) are the Eulerian numbers.
%F E.g.f.: (n - 1)/(n - exp((n-1)*x)) for n = 0 and n >= 2, 1/(1 - x) if n = 1.
%F A(n, 0) = 1; A(n, 1) = n!.
%F A(n, k) = (k - 1)^(n + 1)/k HurwitzLerchPhi(1/k, -n, 0) for k >= 2.
%F A(n, k) = Sum_{j=0..n} j! * Stirling2(n, j) * (k - 1)^(n - j) for k >= 2.
%e Array starts:
%e [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A000012]
%e [1] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... [A000142]
%e [2] 1, 1, 3, 13, 75, 541, 4683, 47293, 545835, ... [A000670]
%e [3] 1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, ... [A122704]
%e [4] 1, 1, 5, 33, 285, 3081, 40005, 606033, 10491885, ... [A255927]
%e [5] 1, 1, 6, 46, 456, 5656, 84336, 1467376, 29175936, ... [A326324]
%e [6] 1, 1, 7, 61, 679, 9445, 158095, 3088765, 68958295, ...
%e [7] 1, 1, 8, 78, 960, 14736, 272448, 5881968, 145105920, ...
%e [8] 1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825, ...
%e Seen as a triangle:
%e [0], 1
%e [1], 1, 1
%e [2], 1, 1, 1
%e [3], 1, 1, 2, 1
%e [4], 1, 1, 3, 6, 1
%e [5], 1, 1, 4, 13, 24, 1
%e [6], 1, 1, 5, 22, 75, 120, 1
%e [7], 1, 1, 6, 33, 160, 541, 720, 1
%e [8], 1, 1, 7, 46, 285, 1456, 4683, 5040, 1
%e [9], 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1
%p A := (n, k) -> add(combinat:-eulerian1(k, j)*n^j, j=0..k):
%p seq(seq(A(n-k, k), k=0..n), n=0..10);
%p # Alternative:
%p egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
%p ser := n -> series(egf(n), x, 21):
%p for n from 0 to 6 do seq(k!*coeff(ser(n), x, k), k=0..9) od;
%t a[n_, 0] := 1; a[n_, 1] := n!;
%t a[n_, k_] := (k - 1)^(n + 1)/k HurwitzLerchPhi[1/k, -n, 0];
%t (* Alternative: *) a[n_, k_] := Sum[StirlingS2[n, j] (k - 1)^(n - j) j!, {j, 0, n}];
%t Table[Print[Table[a[n, k], {n, 0, 10}]], {k, 0, 8}]
%Y Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324.
%K nonn,tabl
%O 0,9
%A _Peter Luschny_, Jun 27 2019
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