cp's OEIS Frontend

A007725 Number of spanning trees of Aztec diamonds of order n.

%I A007725 #54 Feb 28 2023 23:46:15
%S A007725 1,4,768,18170880,48466759778304,14179455913065873408000,
%T A007725 449549878218740179750040371200000,
%U A007725 1534679662450485063038349752542766158611218432,561985025597966566291275288056092110323394467225010519932928
%N A007725 Number of spanning trees of Aztec diamonds of order n.
%H A007725 Timothy Y. Chow, <a href="https://doi.org/10.1090/S0002-9939-97-04049-5">The Q-spectrum and spanning trees of tensor products of bipartite graphs</a>, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3155-3161.
%H A007725 D. E. Knuth, <a href="https://arxiv.org/abs/math/9501234">Aztec Diamonds, Checkerboard Graphs, and Spanning Trees</a>, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.
%H A007725 R. P. Stanley, <a href="https://doi.org/10.1016/S0012-365X(96)83024-X">Spanning trees of Aztec diamonds</a>, Discrete Math. 157 (1996), 375-388 (Problem 251).
%H A007725 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A007725 a(n) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 4^n), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Jan 05 2021
%F A007725 a(n) = 4^(2*n-1) * Product_{1<=j,k<=n-1} (4 - 4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n)))*(4 + 4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n))); [Knuth Eq. (8) p. 3]. - _Seiichi Manyama_, Jan 05 2021
%t A007725 Table[4^n * Product[Product[4 - 4*Cos[j*Pi/(2*n)]*Cos[k*Pi/(2*n)], {k, 1, n-1}], {j, 1, 2*n-1}], {n, 0, 10}] // Round (* _Vaclav Kotesovec_, Jan 05 2021 *)
%o A007725 (PARI) default(realprecision, 120);
%o A007725 {a(n) = if(n==0, 1, round(4^(2*(n-1)*n+1)*prod(j=1, n-1, prod(k=1, n-1, 1-(sin(j*Pi/(2*n))*sin(k*Pi/(2*n)))^2))))} \\ _Seiichi Manyama_, Jan 05 2021
%Y A007725 Cf. A007726, A340166, A340176, A340185, A340352.
%K A007725 nonn
%O A007725 0,2
%A A007725 _Richard Stanley_
%E A007725 More terms from _Alois P. Heinz_, Jan 20 2011
%E A007725 Offset changed (a(0)=1) by _Seiichi Manyama_, Jan 05 2021