A288852 Number T(n,k) of matchings of size k in the n X n X n triangular grid; triangle T(n,k), n>=0, 0<=k<=floor(n*(n+1)/4), read by rows.

1, 1, 1, 3, 1, 9, 15, 2, 1, 18, 99, 193, 108, 6, 1, 30, 333, 1734, 4416, 5193, 2331, 240, 1, 45, 825, 8027, 45261, 151707, 298357, 327237, 180234, 40464, 2238, 1, 63, 1710, 26335, 255123, 1629474, 6995539, 20211423, 38743020, 47768064, 35913207, 15071019

Offset: 0

Author

Alois P. Heinz, Jun 18 2017

Keywords

• nonn
• tabf

Comments

The n X n X n triangular grid has n rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(n) vertices and 3*A000217(n-1) edges altogether.

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1,  3;
  1,  9,  15,    2;
  1, 18,  99,  193,   108,      6;
  1, 30, 333, 1734,  4416,   5193,   2331,    240;
  1, 45, 825, 8027, 45261, 151707, 298357, 327237, 180234, 40464, 2238;
		

Links

• Alois P. Heinz, Rows n = 0..17, flattened
• Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
• Wikipedia, Matching (graph theory)
• Wikipedia, Triangular grid graph

Crossrefs

Columns k=0-1 give: A000012, A045943(n-1) for n>0.

Row sums give A269869.

Last elements of rows give A271610.

Cf. A000217, A011848, A271612, A271614, A271616.

Programs

Maple

b:= proc(l) option remember;  local n, k; n:= nops(l);
      if n=0 then 1
    elif min(l)>0 then b(subsop(-1=NULL, map(h-> h-1, l)))
    else for k to n while l[k]>0 do od; b(subsop(k=1, l))+
         expand(x*(`if`(k1 and l[k-1]=1, b(subsop(k=1, k-1=2, l)), 0)))
      fi
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b([0$n])):
seq(T(n), n=0..10);

Mathematica

b[l_] := b[l] = Module[{n = Length[l], k}, Which[n == 0, 1, Min[l] > 0, b[ReplacePart[l - 1, -1 -> Nothing]], True, For[k = 1, k <= n && l[[k]] > 0, k++]; b[ReplacePart[l, k -> 1]] + x*Expand[If[k < n, b[ReplacePart[l, k -> 2]], 0] + If[k < n && l[[k + 1]] == 0, b[ReplacePart[l, {k -> 1, k + 1 -> 1}]], 0] + If[k > 1 && l[[k - 1]] == 1, b[ReplacePart[l, {k -> 1, k - 1 -> 2}]], 0]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Exponent[#, x]}]&[b[Table[0, n] ]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 24 2018, translated from Maple *)

Formula

T(n,floor(n*(n+1)/4)) = A271610(n).

Sum_{i=0..1} T(n,floor(n*(n+1)/4)-i) = A271612(n).

Sum_{i=0..2} T(n,floor(n*(n+1)/4)-i) = A271614(n).

Sum_{i=0..3} T(n,floor(n*(n+1)/4)-i) = A271616(n).