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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Table of n, a(n) for n=0..48.

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

EXAMPLE

{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}

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

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 *)

CROSSREFS

Cf. A000670, A142075, A173018.

Sequence in context: A133214 A191935 A142075 * A110107 A154537 A201641

Adjacent sequences:  A156362 A156363 A156364 * A156366 A156367 A156368

KEYWORD

nonn,tabf

AUTHOR

Roger L. Bagula, Feb 08 2009

EXTENSIONS

Edited and new name by Peter Luschny, Sep 19 2015

STATUS

approved

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Last modified October 25 00:01 EDT 2020. Contains 338010 sequences. (Running on oeis4.)