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%I #31 Feb 26 2021 07:00:57
%S 4,2304,367500,42467328,4381392020,428652000000,40643137651228,
%T 3771854305099776,344499209234302500,31074298464967845120,
%U 2774871814779003772844,245741556726521856000000,21611621448116558812137652,1889376666754339457990201088,164334311374716912516773437500
%N Row 4 of array in A212796.
%H Seiichi Manyama, <a href="/A212799/b212799.txt">Table of n, a(n) for n = 1..24</a>
%H Germain Kreweras, <a href="https://doi.org/10.1016/0095-8956(78)90021-7">Complexité et circuits Eulériens dans les sommes tensorielles de graphes</a>, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
%F From _Vaclav Kotesovec_, Feb 26 2021: (Start)
%F a(n) ~ (21 + 12*sqrt(3) + 2*sqrt(2*(97 + 56*sqrt(3))))^n * n/4.
%F G.f.: 4*x*(1 + 310*x - 33278*x^2 + 785814*x^3 + 4923451*x^4 - 476492324*x^5 + 8394222196*x^6 - 74272031652*x^7 + 371582629705*x^8 - 981246223862*x^9 + 441533151262*x^10 + 6161037199338*x^11 - 23802532730757*x^12 + 46995963516168*x^13 - 58240430817576*x^14 + 46995963516168*x^15 - 23802532730757*x^16 + 6161037199338*x^17 + 441533151262*x^18 - 981246223862*x^19 + 371582629705*x^20 - 74272031652*x^21 + 8394222196*x^22 - 476492324*x^23 + 4923451*x^24 + 785814*x^25 - 33278*x^26 + 310*x^27 + x^28)/ ((1 - x)^2*(1 - 14*x + x^2)^2*(1 - 6*x + x^2)^2*(1 - 4*x + x^2)^2* (1 - 84*x + 230*x^2 - 84*x^3 + x^4)^2*(1 - 24*x + 50*x^2 - 24*x^3 + x^4)^2). (End)
%t Table[2^(6*n-4)*n*Product[Sin[j*Pi/4]^2 + Sin[k*Pi/n]^2, {j,1,3}, {k,1,n-1}], {n,1,20}]//Round (* _Vaclav Kotesovec_, Feb 26 2021 *)
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o def make_CnXCk(n, k):
%o grids = []
%o for i in range(1, k + 1):
%o for j in range(1, n):
%o grids.append((i + (j - 1) * k, i + j * k))
%o grids.append((i + (n - 1) * k, i))
%o for i in range(1, k * n, k):
%o for j in range(1, k):
%o grids.append((i + j - 1, i + j))
%o grids.append((i + k - 1, i))
%o return grids
%o def A212799(n):
%o if n == 1: return 4
%o if n == 2: return 2304
%o universe = make_CnXCk(4, n)
%o GraphSet.set_universe(universe)
%o spanning_trees = GraphSet.trees(is_spanning=True)
%o return spanning_trees.len()
%o print([A212799(n) for n in range(1, 8)]) # _Seiichi Manyama_, Nov 22 2020
%Y Cf. A003696, A212796, A212798.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, May 27 2012
%E a(10)-a(15) from _Seiichi Manyama_, Nov 22 2020