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

0,4

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.

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

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).

Example

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;

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`(k<n, b(subsop(k=2, l)), 0)+
         `if`(k<n and l[k+1]=0, b(subsop(k=1, k+1=1, l)), 0)+
         `if`(k>1 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 *)

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.

Sequence in context: A141237 A318391 A157399 * A162749 A094796 A056843

Adjacent sequences: A288849 A288850 A288851 * A288853 A288854 A288855

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 18 2017

Status

approved