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 A288852 Number T(n,k) of matchings of size k in the triangle graph of order n; 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The triangle graph of order n 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 Eric Weisstein's World of Mathematics, Triangular Grid Graph Wikipedia, Matching (graph theory) 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`(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 *) 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

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Last modified July 30 03:54 EDT 2021. Contains 346347 sequences. (Running on oeis4.)