A212799 Row 4 of array in A212796.

4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, 3771854305099776, 344499209234302500, 31074298464967845120, 2774871814779003772844, 245741556726521856000000, 21611621448116558812137652, 1889376666754339457990201088, 164334311374716912516773437500

Offset: 1

Author

N. J. A. Sloane, May 27 2012

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..24
• Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.

Crossrefs

Cf. A003696, A212796, A212798.

Programs

Mathematica

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 *)

Python

# Using graphillion
from graphillion import GraphSet
def make_CnXCk(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
        grids.append((i + (n - 1) * k, i))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
        grids.append((i + k - 1, i))
    return grids
def A212799(n):
    if n == 1: return 4
    if n == 2: return 2304
    universe = make_CnXCk(4, n)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A212799(n) for n in range(1, 8)])  # Seiichi Manyama, Nov 22 2020

Formula

From Vaclav Kotesovec, Feb 26 2021: (Start)

a(n) ~ (21 + 12*sqrt(3) + 2*sqrt(2*(97 + 56*sqrt(3))))^n * n/4.

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)

Extensions

a(10)-a(15) from Seiichi Manyama, Nov 22 2020