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A338198 Triangle read by rows, T(n,k) = ((k+1)*2^(n-k)-(k-2)*(-1)^(n-k))/3 for 0 <= k <= n.

%I A338198 #12 Oct 19 2020 16:43:13
%S A338198 1,0,1,2,1,1,2,3,2,1,6,5,4,3,1,10,11,8,5,4,1,22,21,16,11,6,5,1,42,43,
%T A338198 32,21,14,7,6,1,86,85,64,43,26,17,8,7,1,170,171,128,85,54,31,20,9,8,1,
%U A338198 342,341,256,171,106,65,36,23,10,9,1,682,683,512,341,214,127,76,41,26,11,10,1
%N A338198 Triangle read by rows, T(n,k) = ((k+1)*2^(n-k)-(k-2)*(-1)^(n-k))/3 for 0 <= k <= n.
%C A338198 This triangle is related to the Jacobsthal numbers (A001045).
%F A338198 T(n,n) = 1 for n >= 0; T(n,n-1) = n-1 for n > 0.
%F A338198 T(n,k) = T(n-1,k) + 2 * T(n-2,k) for 0 <= k <= n-2.
%F A338198 T(n,k) = 2 * T(n-1,k) - (k-2) * (-1)^(n-k) for 0 <= k < n.
%F A338198 T(n,k) = T(n+1-k,1) + (k-1) * T(n-k,1) for 0 <= k < n.
%F A338198 T(n+1,k) * T(n-1,k) - T(n,k+1) * T(n,k-1) = T(n-k,1)^2 for 0 < k < n.
%F A338198 Row sums are A083579(n+1) for n >= 0.
%F A338198 G.f. of column k >= 0: (1+(k-1)*t) * t^k / (1-t-2*t^2).
%F A338198 G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 - (1+x)*t + 2*x*t^2) / ((1 - x*t)^2 * (1 - t - 2*t^2)).
%F A338198 Conjecture: Let M(n,k) be the matrix inverse of T(n,k), seen as a matrix. Then M(i,j) = 0 if j < 0 or j > i, M(n,n) = 1 for n >= 0, M(n,n-1) = 1-n for n > 0, and M(n,k) = (-1)^(n-k) * (k^2-2) * (n-2)! / k! for 0 <= k <= n-2.
%e A338198 The triangle T(n,k) for 0 <= k <= n starts:
%e A338198 n\k :    0     1     2    3    4    5    6   7   8   9
%e A338198 ======================================================
%e A338198   0 :    1
%e A338198   1 :    0     1
%e A338198   2 :    2     1     1
%e A338198   3 :    2     3     2    1
%e A338198   4 :    6     5     4    3    1
%e A338198   5 :   10    11     8    5    4    1
%e A338198   6 :   22    21    16   11    6    5    1
%e A338198   7 :   42    43    32   21   14    7    6   1
%e A338198   8 :   86    85    64   43   26   17    8   7   1
%e A338198   9 :  170   171   128   85   54   31   20   9   8   1
%e A338198 etc.
%t A338198 Table[((k + 1)*2^(n - k) - (k - 2)*(-1)^(n - k))/3, {n, 0, 11}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Oct 15 2020 *)
%Y A338198 For columns k = 0 to 8 see A078008, A001045, A000079, A001045, A084214, A014551, A083595, A083582, A259713 respectively.
%K A338198 nonn,easy,tabl
%O A338198 0,4
%A A338198 _Werner Schulte_, Oct 15 2020