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A176200 A symmetrical triangle T(n, m) = 2*Eulerian(n+1, m) -1, read by rows.

%I A176200 #12 Sep 08 2022 08:45:52
%S A176200 1,1,1,1,7,1,1,21,21,1,1,51,131,51,1,1,113,603,603,113,1,1,239,2381,
%T A176200 4831,2381,239,1,1,493,8585,31237,31237,8585,493,1,1,1003,29215,
%U A176200 176467,312379,176467,29215,1003,1,1,2025,95679,910383,2620707,2620707,910383,95679,2025,1
%N A176200 A symmetrical triangle T(n, m) = 2*Eulerian(n+1, m) -1, read by rows.
%C A176200 Row sums are: {1, 2, 9, 44, 235, 1434, 10073, 80632, 725751, 7257590, 79833589, ...}.
%H A176200 G. C. Greubel, <a href="/A176200/b176200.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176200 T(n, m) = 2*Eulerian(n+1, m) - 1, where Eulerian(n, k) = A008292(n,k).
%e A176200 Triangle begins as:
%e A176200   1;
%e A176200   1,   1;
%e A176200   1,   7,    1;
%e A176200   1,  21,   21,     1;
%e A176200   1,  51,  131,    51,     1;
%e A176200   1, 113,  603,   603,   113,    1;
%e A176200   1, 239, 2381,  4831,  2381,  239,   1;
%e A176200   1, 493, 8585, 31237, 31237, 8585, 493, 1;
%t A176200 Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j,0,k+1}];
%t A176200 T[n_, m_]:= 2*Eulerian[n+1, m]-1;
%t A176200 Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 25 2019 *)
%o A176200 (PARI) Eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n); {T(n,k) = 2*Eulerian(n+1,k) - 1 };
%o A176200 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Apr 25 2019
%o A176200 (Magma) Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
%o A176200 [[2*Eulerian(n+1,k)-1: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Apr 25 2019
%o A176200 (Sage)
%o A176200 def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o A176200 def T(n,k): return 2*Eulerian(n+1,k)-1
%o A176200 [[T(n,k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Apr 25 2019
%Y A176200 Cf. A008292, A109128.
%K A176200 nonn,tabl
%O A176200 0,5
%A A176200 _Roger L. Bagula_, Apr 11 2010
%E A176200 Edited by _G. C. Greubel_, Apr 25 2019