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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. 0
 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, 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, 342, 341, 256, 171, 106, 65, 36, 23, 10, 9, 1, 682, 683, 512, 341, 214, 127, 76, 41, 26, 11, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS This triangle is related to the Jacobsthal numbers (A001045). LINKS FORMULA T(n,n) = 1 for n >= 0; T(n,n-1) = n-1 for n > 0. T(n,k) = T(n-1,k) + 2 * T(n-2,k) for 0 <= k <= n-2. T(n,k) = 2 * T(n-1,k) - (k-2) * (-1)^(n-k) for 0 <= k < n. T(n,k) = T(n+1-k,1) + (k-1) * T(n-k,1) for 0 <= k < n. 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. Row sums are A083579(n+1) for n >= 0. G.f. of column k >= 0: (1+(k-1)*t) * t^k / (1-t-2*t^2). 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)). 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. EXAMPLE The triangle T(n,k) for 0 <= k <= n starts: n\k :    0     1     2    3    4    5    6   7   8   9 ======================================================   0 :    1   1 :    0     1   2 :    2     1     1   3 :    2     3     2    1   4 :    6     5     4    3    1   5 :   10    11     8    5    4    1   6 :   22    21    16   11    6    5    1   7 :   42    43    32   21   14    7    6   1   8 :   86    85    64   43   26   17    8   7   1   9 :  170   171   128   85   54   31   20   9   8   1 etc. MATHEMATICA 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 *) CROSSREFS For columns k = 0 to 8 see A078008, A001045, A000079, A001045, A084214, A014551, A083595, A083582, A259713 respectively. Sequence in context: A201384 A238348 A143066 * A091598 A144021 A334591 Adjacent sequences:  A338195 A338196 A338197 * A338199 A338200 A338201 KEYWORD nonn,easy,tabl AUTHOR Werner Schulte, Oct 15 2020 STATUS approved

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Last modified August 4 16:25 EDT 2021. Contains 346447 sequences. (Running on oeis4.)