%I #23 Feb 28 2021 08:22:02

%S 10082,401998,19681538,1034315998,55820091938,3044533460992,

%T 166779871224962,9152970837103102,502711247500143362,

%U 27619744381029252622,1517688682641434229698,83401213534557960429502,4583249488240161816039552,251871805990373105011941118,13841645914590329223808310018,760670944425011837491619633038

%N Number of perfect matchings in the graph C_7 X C_{2n}.

%H Seiichi Manyama, <a href="/A230033/b230033.txt">Table of n, a(n) for n = 2..500</a>

%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1016/0031-8914(61)90063-5">The Statistics of Dimers on a Lattice</a>, Physica, 27 (1961), 1209-1225.

%F G.f.: 2*x^2*(5041 -499700*x +20440353*x^2 -466963360*x^3 +6751799885*x^4 -66182756655*x^5 +459438362278*x^6 -2327864968019*x^7 +8797357131438*x^8 -25192378831195*x^9 +55291405473782*x^10 -93750343061691*x^11 +123440474579985*x^12 -126568817064424*x^13 +101127542456783*x^14 -62874205910076*x^15 +30308779015615*x^16 -11259345843608*x^17 +3194422598067*x^18 -683503915153*x^19 +108424368962*x^20 -12458825709*x^21 +1004282914*x^22 -54198917*x^23 +1818498*x^24 -33157*x^25 +239*x^26)/((1 -x)*(1 -13*x +57*x^2 -97*x^3 +57*x^4 -13*x^5 +x^6)*(1 -71*x +952*x^2 -3976*x^3 +6384*x^4 -3976*x^5 +952*x^6 -71*x^7 +x^8)*(1 -54*x +1039*x^2 -9096*x^3 +39037*x^4 -90378*x^5 +118951*x^6 -90378*x^7 +39037*x^8 -9096*x^9 +1039*x^10 -54*x^11 +x^12)).

%F a(n) = sqrt( Product_{j=1..n} Product_{k=1..7} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/7)^2) ). - _Seiichi Manyama_, Feb 14 2021

%o (PARI) default(realprecision, 120);

%o a(n) = round(sqrt(prod(j=1, n, prod(k=1, 7, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/7)^2)))); \\ _Seiichi Manyama_, Feb 14 2021

%Y Column k=7 of A341533.

%Y Cf. A231087, A220864, A231485, A232804, A281583, A308761.

%K nonn

%O 2,1

%A _Sergey Perepechko_, Dec 20 2013