A156365 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.

1, 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

Offset: 0

Author

Roger L. Bagula, Feb 08 2009

Keywords

• nonn
• tabf

Comments

Row sums are the Fubini numbers A000670.

Except for the first term same as A142075. - R. J. Mathar, Feb 19 2009

By the definition of the Eulerian numbers it would be natural to add a 0 at the end of the rows if n > 0. - Peter Luschny, Sep 19 2015

Examples

			Triangle begins:
  1;
  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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A000670, A142075, A173018.

Programs

Magma

Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
A156365:= func< n,k | 2^k*Eulerian(n,k) >;
[1] cat [A156365(n,k): k in [0..n-1], n in [0..12]]; // G. C. Greubel, Jun 05 2021

Maple

A156365 := (n,k) -> combinat:-eulerian1(n,k)*2^k:
for n from 0 to 15 do seq(A156365(n,k), k=0..n) od; # Peter Luschny, Sep 19 2015

Mathematica

(* First program *)
p[x_, n_]= (1-2*x)^(n+1)*PolyLog[-n, 2*x]/(2*x);
Table[CoefficientList[p[x, n], x], {n, 0, 10}]
(* Second program: *)
E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k<0 || k>n = 0; E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
T[0, 0] = 1; T[n_, k_]:= E1[n, k]*2^k;
Table[T[n, k], {n, 0, 10}, {k, 0, Max[0, n-1]}]//Flatten (* Jean-François Alcover, Dec 30 2018, after Peter Luschny *)

Sage

@CachedFunction
def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
def T(n,k): return 2^k*Eulerian(n,k)
[1]+flatten([[T(n,k) for k in (0..n-1)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

Formula

Let p(x,n) = (1 - 2*x)^(n + 1) * Sum_{k>=0} 2^k*(k+1)^n*x^k = (1-2*x)^(1 + n)* polylogarithm(-n, 2*x)/(2*x) then T(n,m) are the coefficients of p(x,n).

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

Extensions

Edited and new name by Peter Luschny, Sep 19 2015