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A142595 Triangle T(n,k) = 2*T(n-1, k-1) + 2*T(n-1, k), read by rows.

%I A142595 #15 Apr 14 2021 06:29:10
%S A142595 1,1,1,1,4,1,1,10,10,1,1,22,40,22,1,1,46,124,124,46,1,1,94,340,496,
%T A142595 340,94,1,1,190,868,1672,1672,868,190,1,1,382,2116,5080,6688,5080,
%U A142595 2116,382,1,1,766,4996,14392,23536,23536,14392,4996,766,1
%N A142595 Triangle T(n,k) = 2*T(n-1, k-1) + 2*T(n-1, k), read by rows.
%C A142595 This triangle is dominated by the Eulerian numbers A008292.
%H A142595 G. C. Greubel, <a href="/A142595/b142595.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A142595 Sum_{k=0..n} T(n, k) =  (4^(n-1) + 2)/3 = A047849(n-1).
%e A142595 Triangle begins as:
%e A142595   1;
%e A142595   1,   1;
%e A142595   1,   4,    1;
%e A142595   1,  10,   10,     1;
%e A142595   1,  22,   40,    22,     1;
%e A142595   1,  46,  124,   124,    46,     1;
%e A142595   1,  94,  340,   496,   340,    94,     1;
%e A142595   1, 190,  868,  1672,  1672,   868,   190,    1;
%e A142595   1, 382, 2116,  5080,  6688,  5080,  2116,  382,   1;
%e A142595   1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1;
%t A142595 T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 2*(T[n-1, k-1] +T[n-1, k])];
%t A142595 Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Apr 13 2021 *)
%t A142595 a[0] = {1}; a[1] = {1, 1};
%t A142595 a[n_]:= a[n]= 2*Join[a[n-1], {-1/2}] + 2*Join[{-1/2}, a[n-1]];
%t A142595 Table[a[n], {n,0,10}]//Flatten (* _Roger L. Bagula_, Dec 09 2008 *)
%o A142595 (Magma)
%o A142595 function T(n,k)
%o A142595   if k eq 1 or k eq n then return 1;
%o A142595   else return 2*(T(n-1, k-1) + T(n-1, k));
%o A142595   end if; return T;
%o A142595 end function;
%o A142595 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 13 2021
%o A142595 (Sage)
%o A142595 @CachedFunction
%o A142595 def T(n,k): return 1 if k==1 or k==n else 2*(T(n-1, k-1) + T(n-1, k))
%o A142595 flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 13 2021
%Y A142595 Cf. A008292, A047849 (row sums), A119258.
%K A142595 nonn,easy,tabl
%O A142595 1,5
%A A142595 _Roger L. Bagula_, Sep 22 2008
%E A142595 Edited by _N. J. A. Sloane_, Dec 11 2008