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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.

%I A156365 #36 Sep 08 2022 08:45:41
%S A156365 1,1,1,2,1,8,4,1,22,44,8,1,52,264,208,16,1,114,1208,2416,912,32,1,240,
%T A156365 4764,19328,19056,3840,64,1,494,17172,124952,249904,137376,15808,128,
%U A156365 1,1004,58432,705872,2499040,2823488,934912,64256,256,1,2026,191360
%N 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.
%C A156365 Row sums are the Fubini numbers A000670.
%C A156365 Except for the first term same as A142075. - _R. J. Mathar_, Feb 19 2009
%C A156365 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
%H A156365 G. C. Greubel, <a href="/A156365/b156365.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A156365 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).
%F A156365 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
%e A156365 Triangle begins:
%e A156365   1;
%e A156365   1;
%e A156365   1,    2;
%e A156365   1,    8,      4;
%e A156365   1,   22,     44,       8;
%e A156365   1,   52,    264,     208,       16;
%e A156365   1,  114,   1208,    2416,      912,       32;
%e A156365   1,  240,   4764,   19328,    19056,     3840,       64;
%e A156365   1,  494,  17172,  124952,   249904,   137376,    15808,     128;
%e A156365   1, 1004,  58432,  705872,  2499040,  2823488,   934912,   64256,    256;
%e A156365   1, 2026, 191360, 3641536, 20965664, 41931328, 29132288, 6123520, 259328, 512;
%p A156365 A156365 := (n,k) -> combinat:-eulerian1(n,k)*2^k:
%p A156365 for n from 0 to 15 do seq(A156365(n,k), k=0..n) od; # _Peter Luschny_, Sep 19 2015
%t A156365 (* First program *)
%t A156365 p[x_, n_]= (1-2*x)^(n+1)*PolyLog[-n, 2*x]/(2*x);
%t A156365 Table[CoefficientList[p[x, n], x], {n, 0, 10}]
%t A156365 (* Second program: *)
%t A156365 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 A156365 T[0, 0] = 1; T[n_, k_]:= E1[n, k]*2^k;
%t A156365 Table[T[n, k], {n, 0, 10}, {k, 0, Max[0, n-1]}]//Flatten (* _Jean-François Alcover_, Dec 30 2018, after _Peter Luschny_ *)
%o A156365 (Magma)
%o A156365 Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
%o A156365 A156365:= func< n,k | 2^k*Eulerian(n,k) >;
%o A156365 [1] cat [A156365(n,k): k in [0..n-1], n in [0..12]]; // _G. C. Greubel_, Jun 05 2021
%o A156365 (Sage)
%o A156365 @CachedFunction
%o A156365 def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o A156365 def T(n,k): return 2^k*Eulerian(n,k)
%o A156365 [1]+flatten([[T(n,k) for k in (0..n-1)] for n in (0..12)]) # _G. C. Greubel_, Jun 05 2021
%Y A156365 Cf. A000670, A142075, A173018.
%K A156365 nonn,tabf
%O A156365 0,4
%A A156365 _Roger L. Bagula_, Feb 08 2009
%E A156365 Edited and new name by _Peter Luschny_, Sep 19 2015