A142075 - OEIS

%I #15 Sep 08 2022 08:45:35

%S 1,1,2,1,8,4,1,22,44,8,1,52,264,208,16,1,114,1208,2416,912,32,1,240,

%T 4764,19328,19056,3840,64,1,494,17172,124952,249904,137376,15808,128,

%U 1,1004,58432,705872,2499040,2823488,934912,64256,256,1,2026,191360,3641536,20965664,41931328,29132288,6123520,259328,512

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

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

%H G. C. Greubel, <a href="/A142075/b142075.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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

%F Sum_{k=1..n} T(n, k) = A000670(n), for n >= 1. - _G. C. Greubel_, Jun 07 2021

%e Triangle begins as:

%e   1;

%e   1,    2;

%e   1,    8,     4;

%e   1,   22,    44,      8;

%e   1,   52,   264,    208,      16;

%e   1,  114,  1208,   2416,     912,      32;

%e   1,  240,  4764,  19328,   19056,    3840,     64;

%e   1,  494, 17172, 124952,  249904,  137376,  15808,   128;

%e   1, 1004, 58432, 705872, 2499040, 2823488, 934912, 64256, 256;

%e   ...

%t (* First program *)

%t p[x_, n_]= (1-2*x)^(n+1)*PolyLog[-n, 2*x]/(2*x);

%t Table[CoefficientList[p[x, n], x], {n,12}]//Flatten

%t (* Second program *)

%t Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];

%t Table[2^(k-1)*Eulerian[n, k-1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 07 2021 *)

%o (Magma)

%o Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;

%o [2^(k-1)*Eulerian(n,k-1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jun 07 2021

%o (Sage)

%o def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))

%o flatten([[2^(k-1)*Eulerian(n,k-1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jun 07 2021

%Y Essentially the same as A156365.

%Y Cf. A000670, A008292, A123125, A173018.

%K nonn,tabl

%O 1,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008

%E Edited and new name by _Joerg Arndt_, Dec 30 2018