cp's OEIS Frontend

A329633 Triangle read by rows: T(n,k) is the number of self-avoiding paths of length n-1+2*k from NW to SW corners in the n X n grid graph (0 <= k <= A000217(n-1), n >= 1).

%I A329633 #42 Dec 10 2023 17:43:02
%S A329633 1,1,1,1,3,5,2,1,6,16,39,61,47,8,1,10,40,125,400,1048,1905,2372,1839,
%T A329633 764,86,1,15,85,335,1237,4638,15860,44365,99815,181995,262414,285086,
%U A329633 218011,104879,26344,1770
%N A329633 Triangle read by rows: T(n,k) is the number of self-avoiding paths of length n-1+2*k from NW to SW corners in the n X n grid graph (0 <= k <= A000217(n-1), n >= 1).
%H A329633 Seiichi Manyama, <a href="/A329633/b329633.txt">Rows n = 1..9, flattened</a>
%F A329633 T(n,0) = 1.
%F A329633 T(n,1) = A000217(n-1).
%e A329633 T(3,0) = 1;
%e A329633    S
%e A329633    |
%e A329633    *
%e A329633    |
%e A329633    E
%e A329633 T(3,1) = 3;
%e A329633    S--*      S--*      S
%e A329633       |         |      |
%e A329633    *--*         *      *--*
%e A329633    |            |         |
%e A329633    E         E--*      E--*
%e A329633 T(3,2) = 5;
%e A329633    S--*--*   S--*--*   S--*--*   S--*      S
%e A329633          |         |         |      |      |
%e A329633    *--*--*      *--*         *      *--*   *--*--*
%e A329633    |            |            |         |         |
%e A329633    E         E--*      E--*--*   E--*--*   E--*--*
%e A329633 T(3,3) = 2;
%e A329633    S--*--*   S  *--*
%e A329633          |   |  |  |
%e A329633    *--*  *   *--*  *
%e A329633    |  |  |         |
%e A329633    E  *--*   E--*--*
%e A329633 Triangle starts:
%e A329633 ==========================================================
%e A329633 n\k| 0   1    2    3     4     5      6 ...    10 ...  15
%e A329633 ---|------------------------------------------------------
%e A329633 1  | 1;
%e A329633 2  | 1,  1;
%e A329633 3  | 1,  3,   5,   2;
%e A329633 4  | 1,  6,  16,  39,   61,   47,     8;
%e A329633 5  | 1, 10,  40, 125,  400, 1048,  1905, ... , 86;
%e A329633 6  | 1, 15,  85, 335, 1237, 4638, 15860, ......... , 1770;
%o A329633 (Python)
%o A329633 # Using graphillion
%o A329633 from graphillion import GraphSet
%o A329633 import graphillion.tutorial as tl
%o A329633 def A329633(n):
%o A329633     if n == 1: return [1]
%o A329633     universe = tl.grid(n - 1, n - 1)
%o A329633     GraphSet.set_universe(universe)
%o A329633     start, goal = 1, n
%o A329633     paths = GraphSet.paths(start, goal)
%o A329633     return [paths.len(n - 1 + 2 * k).len() for k in range(n * (n - 1) // 2 + 1)]
%o A329633 print([i for n in range(1, 7) for i in A329633(n)])
%Y A329633 Row sums give A271507.
%Y A329633 T(n,(n-1)*n/2) gives A000532(n).
%Y A329633 Cf. A000217, A333520.
%K A329633 nonn,tabf
%O A329633 1,5
%A A329633 _Seiichi Manyama_, Mar 30 2020