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%I #39 Apr 29 2023 07:04:50
%S 1,2,6,2196,37004,2317631400,216893681800,2326335506123418128,
%T 1208982377794384163088,2220650888749669503773432361504,
%U 6408743336016148761893699822360672,2015895925780490675949731718780144934779733312,32307672245407537492814937397129549558917000333504
%N Number of perfect matchings in triangle graph with n nodes per side as n runs through numbers congruent to 0 or 3 mod 4.
%D 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).
%H Alois P. Heinz, <a href="/A071093/b071093.txt">Table of n, a(n) for n = 0..44</a>
%H James Propp, <a href="http://arxiv.org/abs/math.CO/9801061">Twenty open problems in enumeration of matchings</a>, arXiv:math/9801061 [math.CO], 1998-1999.
%H James Propp, <a href="http://faculty.uml.edu/jpropp/update.pdf">Updated article</a>
%H James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H James Propp, <a href="http://math.colgate.edu/~integers/x30/x30.pdf">Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds</a>, Integers (2023) Vol. 23, Art. A30.
%F a(2n) = A039907(4n) = A178446(4n), a(2n+1) = A039907(4n+3) = A178446(4n+3). - _Andrew Howroyd_, Mar 06 2016
%p 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
%t 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_ *)
%Y Cf. A039907, A178446, A269869.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E a(9)-a(10) from _Alois P. Heinz_, May 08 2010
%E a(11)-a(12) from _Alois P. Heinz_, Jan 12 2014
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