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

0,5

Links

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

Formula

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

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

Example

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

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.

Sequence in context: A316135 A327620 A325872 * A021896 A188835 A217735

Adjacent sequences: A380366 A380367 A380368 * A380370 A380371 A380372

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 01 2025

Status

approved