A339179 Irregular triangle read by rows: for n >= 2, 2 <= k <= floor(n/2) + 1, T(n,k) = the number of semi-meanders with n top arches, a first arch of length one and k arch groupings.

1, 1, 1, 1, 2, 2, 4, 4, 2, 10, 10, 4, 24, 24, 14, 4, 66, 66, 34, 8, 174, 174, 106, 42, 8, 504, 504, 284, 98, 16, 1406, 1406, 878, 390, 114, 16, 4210, 4210, 2486, 1002, 258, 32, 12198, 12198, 7738, 3652, 1270, 290, 32, 37378, 37378, 22714, 9962, 3140, 642, 64, 111278, 111278, 71370, 34986, 13370, 3794, 706, 64

Offset: 2

Author

Roger Ford, Nov 26 2020

Keywords

• nonn
• tabf

Examples

			For n = 6:   /\ = arch of length one;
       /\             /\             /\             /\
      /  \           //\\           /  \           //\\       4 with 2 groupings
     /  /\\         //  \\         /    \         ///\\\
    /  /  \\       //  /\\\       //\  /\\       ////\\\\
/\ //\//\/\\\, /\ ///\//\\\\, /\ ///\\//\\\, /\ /////\\\\\,
         /\                                              /\
        //\\              /\         /\                 /  \  4 with 3 groupings
       ///\\\       /\   //\\       //\\   /\          //\  \
/\ /\ ////\\\\, /\ //\\ ///\\\, /\ ///\\\ //\\, /\ /\ ///\\/\\,
           /\                                                 2 with 4 groupings
          /  \       /\      /\
/\ /\ /\ //\/\\, /\ //\\ /\ //\\,             T(6,2) = 4, T(6,3) = 4, T(6,4) = 2;
Irregular triangle begins:
    n\k (2) (3) (4) (5) (6)
    2:   1
    3:   1
    4:   1   1
    5:   2   2
    6:   4   4   2
    7:   10  10  4
    8:   24  24  14  4
    9:   66  66  34  8
   10:   174 174 106 42  8
         ...
		

Crossrefs

Cf. A259689, A301620, Row sums: A000682(n-1).

Formula

T(2,2) = T(3,2) = 1.

For n >= 4, T(n,2) = T(n,3) = A000682(n-2).

For n >= 6 and k >= 4, T(n,k) = Sum {x = k-1..floor(n/2)} (A259689(T(n-2,x))).

For n >= 4, A301620(n-3) = Sum {k = 4..floor((n+2)/2)} (T(n,k)).