A326323 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.

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, 33, 160, 541, 720, 1, 1, 1, 7, 46, 285, 1456, 4683, 5040, 1, 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1, 1, 1, 9, 78, 679, 5656, 40005, 202672, 545835, 362880, 1

Offset: 0

Author

Peter Luschny, Jun 27 2019

Keywords

• nonn
• tabl

Examples

			Array starts:
  k=0: 1, 1, 1,  1,    1,     1,      1,        1,         1, ... [A000012]
  k=1: 1, 1, 2,  6,   24,   120,    720,     5040,     40320, ... [A000142]
  k=2: 1, 1, 3, 13,   75,   541,   4683,    47293,    545835, ... [A000670]
  k=3: 1, 1, 4, 22,  160,  1456,  15904,   202672,   2951680, ... [A122704]
  k=4: 1, 1, 5, 33,  285,  3081,  40005,   606033,  10491885, ... [A255927]
  k=5: 1, 1, 6, 46,  456,  5656,  84336,  1467376,  29175936, ... [A326324]
  k=6: 1, 1, 7, 61,  679,  9445, 158095,  3088765,  68958295, ... [A384525]
  k=7: 1, 1, 8, 78,  960, 14736, 272448,  5881968, 145105920, ... [A384514]
  k=8: 1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825, ...
Seen as a triangle:
  [0], 1
  [1], 1, 1
  [2], 1, 1, 1
  [3], 1, 1, 2,  1
  [4], 1, 1, 3,  6,   1
  [5], 1, 1, 4, 13,  24,    1
  [6], 1, 1, 5, 22,  75,  120,     1
  [7], 1, 1, 6, 33, 160,  541,   720,     1
  [8], 1, 1, 7, 46, 285, 1456,  4683,  5040,     1
  [9], 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1
		

Links

• OEIS Wiki, Eulerian polynomials.

Crossrefs

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A384525, A384514.

Programs

Maple

A := (n, k) -> add(combinat:-eulerian1(k, j)*n^j, j=0..k):
seq(seq(A(n-k, k), k=0..n), n=0..10);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n  - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(k!*coeff(ser(n), x, k), k=0..9) od;

Mathematica

a[n_, 0] := 1; a[n_, 1] := n!;
a[n_, k_] := (k - 1)^(n + 1)/k HurwitzLerchPhi[1/k, -n, 0];
(* Alternative: *) a[n_, k_] := Sum[StirlingS2[n, j] (k - 1)^(n - j) j!, {j, 0, n}];
Table[Print[Table[a[n, k], {n, 0, 10}]], {k, 0, 8}]

Formula

A(n, k) = Sum_{j=0..k} a(k, j)*n^j where a(k, j) are the Eulerian numbers.

E.g.f.: (n - 1)/(n - exp((n-1)*x)) for n = 0 and n >= 2, 1/(1 - x) if n = 1.

A(n, 0) = 1; A(n, 1) = n!.

A(n, k) = (k - 1)^(n + 1)/k HurwitzLerchPhi(1/k, -n, 0) for k >= 2.

A(n, k) = Sum_{j=0..n} j! * Stirling2(n, j) * (k - 1)^(n - j) for k >= 2.