A142075 Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.

1, 1, 2, 1, 8, 4, 1, 22, 44, 8, 1, 52, 264, 208, 16, 1, 114, 1208, 2416, 912, 32, 1, 240, 4764, 19328, 19056, 3840, 64, 1, 494, 17172, 124952, 249904, 137376, 15808, 128, 1, 1004, 58432, 705872, 2499040, 2823488, 934912, 64256, 256, 1, 2026, 191360, 3641536, 20965664, 41931328, 29132288, 6123520, 259328, 512

Offset

1,3

Comments

Same as A156365, except for the additional a(0) = 1 there.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

G.f.: 1/x/Q(0) -1/x, where Q(k) = 1 - x*(k+1)/( 1 - y*2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2013

Sum_{k=1..n} T(n, k) = A000670(n), for n >= 1. - G. C. Greubel, Jun 07 2021

Example

Triangle begins as:
  1;
  1,    2;
  1,    8,     4;
  1,   22,    44,      8;
  1,   52,   264,    208,      16;
  1,  114,  1208,   2416,     912,      32;
  1,  240,  4764,  19328,   19056,    3840,     64;
  1,  494, 17172, 124952,  249904,  137376,  15808,   128;
  1, 1004, 58432, 705872, 2499040, 2823488, 934912, 64256, 256;
  ...

Mathematica

(* First program *)
p[x_, n_]= (1-2*x)^(n+1)*PolyLog[-n, 2*x]/(2*x);
Table[CoefficientList[p[x, n], x], {n, 12}]//Flatten
(* Second program *)
Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
Table[2^(k-1)*Eulerian[n, k-1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)

Prog

(Magma)
Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[2^(k-1)*Eulerian(n, k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jun 07 2021
(Sage)
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1))
flatten([[2^(k-1)*Eulerian(n, k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 07 2021

Crossrefs

Essentially the same as A156365.

Cf. A000670, A008292, A123125, A173018.

Sequence in context: A133214 A191935 A156365 * A366173 A110107 A154537

Adjacent sequences: A142072 A142073 A142074 * A142076 A142077 A142078

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Extensions

Edited and new name by Joerg Arndt, Dec 30 2018

Status

approved