A272098 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.

1, 2, 0, 4, 1, 0, 8, 7, 1, 0, 16, 33, 15, 1, 0, 32, 131, 131, 31, 1, 0, 64, 473, 883, 473, 63, 1, 0, 128, 1611, 5111, 5111, 1611, 127, 1, 0, 256, 5281, 26799, 44929, 26799, 5281, 255, 1, 0, 512, 16867, 131275, 344551, 344551, 131275, 16867, 511, 1, 0

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Peter Luschny, Extensions of the binomial

Formula

E.g.f.: (exp(x)*(y - 1))/(y - exp(x*(y - 1))). - Peter Luschny, Aug 14 2022

T(n,k) = Sum_{i=0..n} Binomial(n,i)*Eulerian(i,k), where Eulerian(n,k) = Eulerian numbers A173018. Equivalently, if T is the matrix generated by T(n,k), B is the binomial matrix and E is the Eulerian matrix, then T = B E. - Emanuele Munarini, Oct 19 2023

Example

Triangle starts:
  [1]
  [2, 0]
  [4, 1, 0]
  [8, 7, 1, 0]
  [16, 33, 15, 1, 0]
  [32, 131, 131, 31, 1, 0]
  [64, 473, 883, 473, 63, 1, 0]
  [128, 1611, 5111, 5111, 1611, 127, 1, 0]

Maple

T := (n, k) -> add((-1)^(n-j)*combinat:-eulerian1(j, k)*binomial(-j-1, -n-1), j=0..n): seq(seq(T(n, k), k=0..n), n=0..10);
# Or:
egf := (exp(x)*(y - 1))/(y - exp(x*(y - 1))); ser := series(egf, x, 12):
cx := n -> series(coeff(ser, x, n), y, n + 2):
seq(seq(n!*coeff(cx(n), y, k), k = 0..n), n = 0..9); # Peter Luschny, Aug 14 2022

Mathematica

<<Combinatorica`
Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1, -n-1] Eulerian[j, k], {j, 0, n}], {n, 0, 9}, {k, 0, n}]]
Flatten[Table[Sum[Binomial[n, i] Eulerian[i, k], {i, 0, n}], {n, 0, 12}, {k, 0, n}]] (* Emanuele Munarini, Oct 19 2023 *)

Crossrefs

Cf. A000522 (row sums), A000079 (col. 0), A066810 (col. 1).

Cf. A173018.

Sequence in context: A153342 A144258 A056859 * A291929 A327807 A188448

Adjacent sequences: A272095 A272096 A272097 * A272099 A272100 A272101

Keyword

nonn,tabl

Author

Peter Luschny, Apr 20 2016

Status

approved