A039907 Number of perfect matchings in triangle graph with n nodes per side.

1, 0, 0, 2, 6, 0, 0, 2196, 37004, 0, 0, 2317631400, 216893681800, 0, 0, 2326335506123418128, 1208982377794384163088, 0, 0, 2220650888749669503773432361504, 6408743336016148761893699822360672, 0, 0, 2015895925780490675949731718780144934779733312

Offset

0,4

References

J. 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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

J. Propp, Twenty open problems in enumeration of matchings, arXiv:math/9801061 [math.CO], 1998-1999.

J. Propp, Updated article

J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Formula

a(n) = A178446(n) if A000217(n) is even, a(n) = 0 otherwise. - Andrew Howroyd, Mar 06 2016

Maple

with(LinearAlgebra): a:= 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: seq(a(n), n=0..20); # Alois P. Heinz, May 08 2010

Mathematica

a[n_] := a[n] = Module[{l, ll, i, j, h0, h1, M}, If[n == 0 , Return[1]]; If[n < 0 || MemberQ[{1, 2}, Mod[n, 4]], 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++, l = Join[l, {h1, h1+1}]; If[Mod [i, 2] == 1, l = Join[l, {h1, h0}]; h1 = h1+1; l = Join[l, {h1, h0}]; h0 = h0+1, l = Join[l, {h0, h1}]; h1 = h1+1; l = Join[l, {h0, h1}]; h0 = h0+1]]]; M[_, _] = 0; Do[M[ll[[1]], ll[[2]]] = 1; M[ll[[2]], ll[[1]]] = -1, {ll, Partition[l, 2]}]; Sqrt[Det[Array[M, {n*(n+1)/2, n*(n+1)/2}]]]]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 17 2014, after Alois P. Heinz *)

Crossrefs

Cf. A071093, A269869, A178446.

Sequence in context: A186502 A165733 A330327 * A072340 A118354 A080730

Adjacent sequences: A039904 A039905 A039906 * A039908 A039909 A039910

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

a(17)-a(20) from Alois P. Heinz, May 08 2010

a(21)-a(23) from Alois P. Heinz, Jan 12 2014

Status

approved