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

1,3

Links

Table of n, a(n) for n=1..55.

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

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)

Example

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
...

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 *)

Crossrefs

Cf. A008292, A211232-A211235.

Sequence in context: A210229 A210213 A305695 * A134626 A347633 A115450

Adjacent sequences: A211232 A211233 A211234 * A211236 A211237 A211238

Keyword

nonn,tabl

Author

N. J. A. Sloane, Apr 05 2012

Extensions

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

Status

approved