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A177696 Symmetrical triangle read by rows: T(n, k) = m*(T(n-1, k-1) + T(n-1, k)), where T(n, 1) = T(n, n) = n, and m = 2.

%I A177696 #7 Oct 02 2024 06:48:28
%S A177696 1,2,2,3,8,3,4,22,22,4,5,52,88,52,5,6,114,280,280,114,6,7,240,788,
%T A177696 1120,788,240,7,8,494,2056,3816,3816,2056,494,8,9,1004,5100,11744,
%U A177696 15264,11744,5100,1004,9,10,2026,12208,33688,54016,54016,33688,12208,2026,10
%N A177696 Symmetrical triangle read by rows: T(n, k) = m*(T(n-1, k-1) + T(n-1, k)), where T(n, 1) = T(n, n) = n, and m = 2.
%H A177696 G. C. Greubel, <a href="/A177696/b177696.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A177696 T(n, k) = m*(T(n-1, k-1) + T(n-1, k)), where T(n, 1) = T(n, n) = n, and m = 2.
%F A177696 From _G. C. Greubel_, Oct 02 2024: (Start)
%F A177696 Sum_{k=1..n} T(n, k) = (1/9)*(7*4^n + 6*n + 2).
%F A177696 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1-(-1)^n)*(2-n) - [n=1]. (End)
%e A177696 Triangle begins as:
%e A177696    1;
%e A177696    2,    2;
%e A177696    3,    8,     3;
%e A177696    4,   22,    22,     4;
%e A177696    5,   52,    88,    52,     5;
%e A177696    6,  114,   280,   280,   114,     6;
%e A177696    7,  240,   788,  1120,   788,   240,     7;
%e A177696    8,  494,  2056,  3816,  3816,  2056,   494,     8;
%e A177696    9, 1004,  5100, 11744, 15264, 11744,  5100,  1004,    9;
%e A177696   10, 2026, 12208, 33688, 54016, 54016, 33688, 12208, 2026, 10;
%t A177696 m = 2;T[n_, k_]:= T[n, k]= If[k==1 || k==n, n, m*(T[n-1, k-1] + T[n-1, k])];Table[T[n, k], {n, 10}, {k, n}]//Flatten
%o A177696 (Magma)
%o A177696 function T(n, k) // T = A177696
%o A177696   if k lt 1 or k gt n then return 0;
%o A177696   elif k eq 1 or k eq n then return n;
%o A177696   else return 2*(T(n-1, k-1) + T(n-1, k));
%o A177696   end if;
%o A177696 end function;
%o A177696 [T(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 02 2024
%o A177696 (SageMath)
%o A177696 @CachedFunction
%o A177696 def T(n, k): # T = A177696
%o A177696     if (k<0 or k>n): return 0
%o A177696     elif (k==1 or k==n): return n
%o A177696     else: return 2*(T(n-1, k-1) + T(n-1, k))
%o A177696 flatten([[T(n, k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 02 2024
%Y A177696 Cf. A051597 (m=1).
%K A177696 nonn,tabl
%O A177696 1,2
%A A177696 _Roger L. Bagula_, May 11 2010
%E A177696 Edited by _G. C. Greubel_, Oct 02 2024