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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

Table of n, a(n) for n=0..65.

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<k<n := row[n][[k+1]];

Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Aug 07 2018 *)

CROSSREFS

A001006, Cf. A086246

Sequence in context: A232084 A261359 A217680 * A098691 A035364 A261734

Adjacent sequences:  A144215 A144216 A144217 * A144219 A144220 A144221

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 14 2008

STATUS

approved

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Last modified October 19 02:23 EDT 2018. Contains 316327 sequences. (Running on oeis4.)