A006660 -id:A006660 - cp's OEIS frontend

A259974 Irregular triangle read by rows: T(n,k) = number of meanders with n bridges in which the first bridge is bridge k.

1, 1, 1, 1, 2, 1, 3, 2, 3, 8, 3, 3, 14, 7, 7, 14, 42, 14, 11, 14, 81, 36, 28, 36, 81, 262, 81, 57, 57, 81, 538, 221, 155, 155, 221, 538, 1828, 538, 353, 316, 353, 538, 3926, 1530, 1003, 902, 1003, 1530, 3926, 13820, 3926, 2458, 2053, 2053, 2458, 3926, 30694, 11510, 7214, 6059, 6059, 7214, 11510, 30694, 110954, 30694, 18575, 14810, 13827, 14810, 18575, 30694

Offset: 1

N. J. A. Sloane, Jul 12 2015

			Triangle begins:
1,
1,
1,1,
2,1,
3,2,3,
8,3,3,
14,7,7,14,
42,14,11,14,
81,36,28,36,81,
262,81,57,57,81,
538,221,155,155,221,538,
...

Lando, S. K. and Zvonkin, A. K. Plane and projective meanders. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).
Lando, S. K. and Zvonkin, A. K. Meanders. In Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 117-144.

Andrew Howroyd, Table of n, a(n) for n = 1..420
S. K. Lando and A. K. Zvonkin , Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117 (1993), no. 1-2, 227-241.

Diagonals are A005316, A006660, A006661, A006662. Cf. A008828.

T(12,k)-T(40,k) from Andrew Howroyd, Dec 15 2015

A380369 Triangle read by rows: T(n,k) is the number of open meanders with 2n crossings and k exterior top arches, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 7, 6, 1, 0, 36, 32, 12, 1, 0, 221, 202, 94, 20, 1, 0, 1530, 1417, 728, 220, 30, 1, 0, 11510, 10752, 5854, 2090, 445, 42, 1, 0, 92114, 86554, 48942, 19300, 5160, 812, 56, 1, 0, 773259, 729716, 423778, 178478, 54758, 11396, 1372, 72, 1, 0, 6743122, 6384353, 3781926, 1669062, 561514, 138866, 23072, 2184, 90, 1

Offset: 0

Andrew Howroyd, Feb 01 2025

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,     7,     6,     1;
  0,    36,    32,    12,     1;
  0,   221,   202,    94,    20,    1;
  0,  1530,  1417,   728,   220,   30,   1;
  0, 11510, 10752,  5854,  2090,  445,  42,  1;
  0, 92114, 86554, 48942, 19300, 5160, 812, 56, 1;
  ...
The T(2,1) = 2 open meanders are:
         __           __
        /  \         /  \
   ... / /\ \..  .. / /\ \ ...
      / /  \/       \/  \ \
The T(2,2) = 1 open meander is:
   ... /\../\ ...
      /  \/  \

Andrew Howroyd, Table of n, a(n) for n = 0..230

Row sums are A077054.
Main diagonal is A000012.
Second diagonal is A002378.
Cf. A005316, A006660 (bisection gives column 1), A077056 (total number of exterior top arches), A259689 (for semi-meanders), A259974.

A077056(n) = Sum_{k=1..n} k*T(n,k).

T(n,1) = A006660(2*n + 1).

cp's OEIS Frontend

A259974 Irregular triangle read by rows: T(n,k) = number of meanders with n bridges in which the first bridge is bridge k.

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