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 A144218 Eigentriangle, row sums and borders = offset variations of Motzkin numbers 3
 1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 4, 2, 2, 4, 9, 9, 4, 4, 4, 9, 21, 21, 9, 8, 8, 9, 21, 51, 51, 21, 18, 16, 18, 21, 51, 127, 127, 51, 42, 36, 36, 42, 51, 127, 323, 323, 127, 102, 84, 81, 84, 102, 127, 323, 835, 835, 323, 254, 204, 189, 189, 204, 254, 323, 835, 2188 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Right border = Motzkin numbers, A001006: (1, 1, 2, 4, 9, 21,...). Row sums = (1, 2, 4, 9, 21,...); Left border = A086246: (1, 1, 1, 2, 4, 9, 21,...).Q Sum of n-th row terms = rightmost term of next row. LINKS P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011. FORMULA Let A = an infinite lower triangular matrix with A086246: (1, 1, 1, 2, 4, 9, 21, 51,...) in every column; and B = an infinite lower triangular matrix with A001006, (1, 1, 2, 4, 9, 21,...) as the main diagonal and the rest zeros. a144218 = A*B. EXAMPLE First few rows of the triangle = 1; 1, 1; 1, 1, 2; 2, 1, 2, 4; 4, 2, 2, 4, 9; 9, 4, 4, 4, 9, 21; 21, 9, 8, 8, 9, 21, 51; 51, 21, 18, 16, 18, 21, 51, 127; 127, 51, 42, 36, 36, 42, 51, 127, 323; 323, 127, 102, 84, 81, 84, 102, 127, 323, 835; 835, 323, 254, 204, 189, 189, 204, 254, 835, 2188; ... Row 4 = (4, 2, 2, 4, 9) = termwise products of (4, 2, 1, 1, 1) and (1, 1, 2, 4, 9) = (4*1, 2*1, 1*2, 1*4, 1*9). MATHEMATICA nmax = 10; T[0, 0] = T[1, 0] = 1; T[n_, 0]  := Hypergeometric2F1[3/2, 1-n, 3, 4] // Abs; T[n_, n_] := Hypergeometric2F1[(1-n)/2, -n/2, 2, 4]; row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ]; T[n_, k_] /; 0

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Last modified November 12 12:43 EST 2018. Contains 317109 sequences. (Running on oeis4.)