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%I #27 Aug 01 2023 14:57:19
%S 1,1,1,3,1,9,15,2,1,18,99,193,108,6,1,30,333,1734,4416,5193,2331,240,
%T 1,45,825,8027,45261,151707,298357,327237,180234,40464,2238,1,63,1710,
%U 26335,255123,1629474,6995539,20211423,38743020,47768064,35913207,15071019
%N 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.
%C 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.
%H Alois P. Heinz, <a href="/A288852/b288852.txt">Rows n = 0..17, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Matching_(graph_theory)">Matching (graph theory)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%F T(n,floor(n*(n+1)/4)) = A271610(n).
%F Sum_{i=0..1} T(n,floor(n*(n+1)/4)-i) = A271612(n).
%F Sum_{i=0..2} T(n,floor(n*(n+1)/4)-i) = A271614(n).
%F Sum_{i=0..3} T(n,floor(n*(n+1)/4)-i) = A271616(n).
%e Triangle T(n,k) begins:
%e 1;
%e 1;
%e 1, 3;
%e 1, 9, 15, 2;
%e 1, 18, 99, 193, 108, 6;
%e 1, 30, 333, 1734, 4416, 5193, 2331, 240;
%e 1, 45, 825, 8027, 45261, 151707, 298357, 327237, 180234, 40464, 2238;
%p b:= proc(l) option remember; local n, k; n:= nops(l);
%p if n=0 then 1
%p elif min(l)>0 then b(subsop(-1=NULL, map(h-> h-1, l)))
%p else for k to n while l[k]>0 do od; b(subsop(k=1, l))+
%p expand(x*(`if`(k<n, b(subsop(k=2, l)), 0)+
%p `if`(k<n and l[k+1]=0, b(subsop(k=1, k+1=1, l)), 0)+
%p `if`(k>1 and l[k-1]=1, b(subsop(k=1, k-1=2, l)), 0)))
%p fi
%p end:
%p T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b([0$n])):
%p seq(T(n), n=0..10);
%t 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 T[n_] := Table[Coefficient[#, x, i], {i, 0, Exponent[#, x]}]&[b[Table[0, n] ]];
%t Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, May 24 2018, translated from Maple *)
%Y Columns k=0-1 give: A000012, A045943(n-1) for n>0.
%Y Row sums give A269869.
%Y Last elements of rows give A271610.
%Y Cf. A000217, A011848, A271612, A271614, A271616.
%K nonn,tabf
%O 0,4
%A _Alois P. Heinz_, Jun 18 2017
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