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A071093
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Number of perfect matchings in triangle graph with n nodes per side as n runs through numbers congruent to 0 or 3 mod 4.
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3
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1, 2, 6, 2196, 37004, 2317631400, 216893681800, 2326335506123418128, 1208982377794384163088, 2220650888749669503773432361504, 6408743336016148761893699822360672, 2015895925780490675949731718780144934779733312, 32307672245407537492814937397129549558917000333504
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OFFSET
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0,2
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REFERENCES
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James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 17).
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LINKS
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FORMULA
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MAPLE
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with(LinearAlgebra): b:= proc(n) option remember; local l, ll, i, j, h0, h1, M; if n=0 then return 1 fi; if n<0 or member(irem(n, 4), [1, 2]) then return 0 fi; l:= []; for j from 1 to n-1 do h0:= j*(j-1)/2+1; h1:= j*(j+1)/2+1; for i from 1 to j do l:= [l[], [h1, h1+1]]; if irem(i, 2)=1 then l:= [l[], [h1, h0]]; h1:= h1+1; l:=[l[], [h1, h0]]; h0:=h0+1 else l:= [l[], [h0, h1]]; h1:= h1+1; l:=[l[], [h0, h1]]; h0:=h0+1 fi od od; M:= Matrix((n+1)*n/2); for ll in l do M[ll[1], ll[2]]:= 1; M[ll[2], ll[1]]:= -1 od: isqrt(Determinant(M)); end: a:= n-> b(2*n +irem(n, 2)): seq(a(n), n=0..10); # Alois P. Heinz, May 08 2010
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MATHEMATICA
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b[n_] := b[n] = Module[{l, ll, i, j, h0, h1, M}, If[n == 0 , Return[1]]; If[n<0 || MatchQ[Mod[n, 4], 1|2] , Return[0]]; l = {}; For[j = 1, j <= n-1, j++, h0 = j*(j-1)/2+1; h1 = j*(j+1)/2+1; For[i = 1, i <= j, i++, AppendTo[l, {h1, h1+1}]; If[Mod[i, 2] == 1, AppendTo[l, {h1, h0}]; h1++; AppendTo[l, {h1, h0}]; h0++ , AppendTo[l, {h0, h1}]; h1++; AppendTo[l, {h0, h1}]; h0++ ]]]; M[_, _] = 0; (M[#[[1]], #[[2]]] = 1; M[#[[2]], #[[1]]] = -1)& /@ l; Sqrt[Det[Array[M, {n*(n+1)/2, n*(n+1)/2}]]]]; a[n_] := b[2*n + Mod[n, 2]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 29 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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