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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.
6
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
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
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.
Sequence in context: A141237 A318391 A157399 * A162749 A094796 A056843
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 18 2017
STATUS
approved