A211235 Array of generalized Eulerian numbers C(n,k) read by antidiagonals.

1, 1, 2, 1, 4, 3, 1, 7, 10, 4, 1, 12, 27, 20, 5, 1, 21, 69, 77, 35, 6, 1, 38, 176, 272, 182, 56, 7, 1, 71, 456, 936, 846, 378, 84, 8, 1, 136, 1205, 3210, 3750, 2232, 714, 120, 9, 1, 265, 3247, 11075, 16290, 12342, 5214, 1254, 165, 10

Offset: 1

Author

N. J. A. Sloane, Apr 05 2012

Keywords

• nonn
• tabl

Examples

			Array begins:
  1,       2,   3,   4,    5,     6, ... A000027
  1,       4,  10,  20,   35,    56, ... A000292
  1,       7,  27,  77,  182,   378, ... A005585
  1,      12,  69, 272,  846,  2232, ... A101097
  1,      21, 176, 936, 3750, 12342, ... A254681
     A005126,
  ...
Triangle begins:
  1
  1  2
  1  4   3
  1  7  10   4
  1 12  27  20   5
  1 21  69  77  35  6
  1 38 176 272 182 56 7
  ...
		

Links

• D. H. Lehmer, Generalized Eulerian numbers, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).

Crossrefs

Cf. A008292, A211232-A211235.

Programs

Maple

A211235 := (n, k) -> add(binomial(n-i, k-i)*i^(n-k), i = 1 .. k): for n from 1 to 10 do seq(A211235(n, k), k = 1 .. n) end do; # Peter Bala, Oct 27 2015

Mathematica

T[n_, k_] := Sum[Binomial[n-i, k-i] * i^(n-k), {i, 1, k}]; Table[T[n, k], {n,1,10}, {k,1,n}] //Flatten (* Amiram Eldar, Nov 30 2018 *)

Formula

From Peter Bala, Oct 27 2015: (Start)

O.g.f. of n-th row of square array: 1/(1 - x)^n * (x*d/dx)^n log(1/(1 - x)), for n >= 1.

E.g.f. of square array: log((1 - x)/(1 - x*exp(t/(1 - x)))).

Read as a triangle: T(n,k) = Sum_{i = 1..k} binomial(n-i,k-i)*i^(n-k) for 1 <= k <= n.

n-th row polynomial of triangle: Sum_{i = 0..n-1} x^i*(x + i)^(n-i). (End)

Extensions

Terms a(37)-a(55) added by Peter Bala, Oct 27 2015