%I #20 Feb 01 2024 08:21:16

%S 1,1,1,1,1,1,1,2,1,1,1,6,3,1,1,1,24,13,4,1,1,1,120,75,22,5,1,1,1,720,

%T 541,160,33,6,1,1,1,5040,4683,1456,285,46,7,1,1,1,40320,47293,15904,

%U 3081,456,61,8,1,1,1,362880,545835,202672,40005,5656,679,78,9,1,1

%N A(n, k) = Sum_{j=0..n} j!*Stirling2(n, j)*(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

%H Paolo Xausa, <a href="/A332700/b332700.txt">Table of n, a(n) for n = 0..11475</a> (antidiagonals 0..150 of the square, flattened).

%F A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).

%F A(n, 1) = n!*[x^n] 1/(1-x).

%F A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.

%F A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.

%e Array begins:

%e [0] 1, 1, 1, 1, 1, 1, 1, ... A000012

%e [1] 1, 1, 1, 1, 1, 1, 1, ... A000012

%e [2] 1, 2, 3, 4, 5, 6, 7, ... A000027

%e [3] 1, 6, 13, 22, 33, 46, 61, ... A028872

%e [4] 1, 24, 75, 160, 285, 456, 679, ...

%e [5] 1, 120, 541, 1456, 3081, 5656, 9445, ...

%e [6] 1, 720, 4683, 15904, 40005, 84336, 158095, ...

%e [7] 1, 5040, 47293, 202672, 606033, 1467376, 3088765, ...

%e [8] 1, 40320, 545835, 2951680, 10491885, 29175936, 68958295, ...

%e [9] 1, 362880, 7087261, 48361216, 204343641, 652606336, 1731875605, ...

%e A000142, A000670, A122704, A255927, A326324, ...

%e Seen as a triangle:

%e [0] [1]

%e [1] [1, 1]

%e [2] [1, 1, 1]

%e [3] [1, 2, 1, 1]

%e [4] [1, 6, 3, 1, 1]

%e [5] [1, 24, 13, 4, 1, 1]

%e [6] [1, 120, 75, 22, 5, 1, 1]

%e [7] [1, 720, 541, 160, 33, 6, 1, 1]

%e [8] [1, 5040, 4683, 1456, 285, 46, 7, 1, 1]

%e [9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]

%p # Prints array by row.

%p A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):

%p seq(print(seq(A(n,k), k=0..10)), n=0..8);

%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(n!*coeff(ser(k), x, n), k=0..9) od;

%p # Or:

%p A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else

%p polylog(-n, 1/k)*(k-1)^(n+1)/k fi:

%p for n from 0 to 6 do seq(A(n, k), k=0..9) od;

%t A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];

%t Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Feb 01 2024 *)

%o (Sage)

%o def T(n, k):

%o return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))

%o for n in range(8): print([T(n, k) for k in range(8)])

%Y The matrix transpose of A326323.

%Y Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Feb 28 2020