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A142596 Triangle T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.

%I A142596 #8 Apr 14 2021 06:29:19
%S A142596 1,1,1,1,6,1,1,21,21,1,1,66,126,66,1,1,201,576,576,201,1,1,606,2331,
%T A142596 3456,2331,606,1,1,1821,8811,17361,17361,8811,1821,1,1,5466,31896,
%U A142596 78516,104166,78516,31896,5466,1,1,16401,112086,331236,548046,548046,331236,112086,16401,1
%N A142596 Triangle T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.
%H A142596 G. C. Greubel, <a href="/A142596/b142596.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A142596 T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1.
%F A142596 Sum_{k=1..n} T(n, k) = (6^(n-1) + 4)/5 = A047851(n-1). - _G. C. Greubel_, Apr 13 2021
%e A142596 The triangle begins as:
%e A142596   1;
%e A142596   1,     1;
%e A142596   1,     6,      1;
%e A142596   1,    21,     21,      1;
%e A142596   1,    66,    126,     66,      1;
%e A142596   1,   201,    576,    576,    201,      1;
%e A142596   1,   606,   2331,   3456,   2331,    606,      1;
%e A142596   1,  1821,   8811,  17361,  17361,   8811,   1821,      1;
%e A142596   1,  5466,  31896,  78516, 104166,  78516,  31896,   5466,     1;
%e A142596   1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1;
%t A142596 T[n_, k_]:= T[n,k]= If[k==1 || k==n, 1, T[n-1, k-1] +3*T[n-1, k] +2*T[n-1, k-1]];
%t A142596 Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Apr 13 2021 *)
%o A142596 (Magma)
%o A142596 function T(n,k)
%o A142596   if k eq 1 or k eq n then return 1;
%o A142596   else return T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1);
%o A142596   end if; return T;
%o A142596 end function;
%o A142596 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 13 2021
%o A142596 (Sage)
%o A142596 @CachedFunction
%o A142596 def T(n,k): return 1 if k==1 or k==n else T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1)
%o A142596 flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 13 2021
%Y A142596 Cf. A008292, A047851, A060187, A119258.
%K A142596 nonn,tabl
%O A142596 1,5
%A A142596 _Roger L. Bagula_, Sep 22 2008
%E A142596 Edited by _G. C. Greubel_, Apr 13 2021