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 A144218 Eigentriangle, row sums and borders = offset variations of Motzkin numbers 3

%I

%S 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,

%T 16,18,21,51,127,127,51,42,36,36,42,51,127,323,323,127,102,84,81,84,

%U 102,127,323,835,835,323,254,204,189,189,204,254,323,835,2188

%N Eigentriangle, row sums and borders = offset variations of Motzkin numbers

%C Right border = Motzkin numbers, A001006: (1, 1, 2, 4, 9, 21,...).

%C Row sums = (1, 2, 4, 9, 21,...);

%C Left border = A086246: (1, 1, 1, 2, 4, 9, 21,...).Q Sum of n-th row terms = rightmost term of next row.

%H P. Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

%F 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.

%F a144218 = A*B.

%e First few rows of the triangle =

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 2, 1, 2, 4;

%e 4, 2, 2, 4, 9;

%e 9, 4, 4, 4, 9, 21;

%e 21, 9, 8, 8, 9, 21, 51;

%e 51, 21, 18, 16, 18, 21, 51, 127;

%e 127, 51, 42, 36, 36, 42, 51, 127, 323;

%e 323, 127, 102, 84, 81, 84, 102, 127, 323, 835;

%e 835, 323, 254, 204, 189, 189, 204, 254, 835, 2188;

%e ...

%e 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).

%t nmax = 10;

%t T[0, 0] = T[1, 0] = 1;

%t T[n_, 0] := Hypergeometric2F1[3/2, 1-n, 3, 4] // Abs;

%t T[n_, n_] := Hypergeometric2F1[(1-n)/2, -n/2, 2, 4];

%t row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ];

%t T[n_, k_] /; 0<k<n := row[n][[k+1]];

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)

%Y A001006, Cf. A086246

%K nonn,tabl

%O 0,6

%A _Gary W. Adamson_, Sep 14 2008

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Last modified October 16 00:50 EDT 2018. Contains 316252 sequences. (Running on oeis4.)