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.

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

Author

Andrew Howroyd, Feb 01 2025

Keywords

• nonn
• tabl

Examples

			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:
   ... /\../\ ...
      /  \/  \
		

Links

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

Crossrefs

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.

Formula

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

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