A212798 Row 3 of array in A212796.

3, 294, 11664, 367500, 10609215, 292626432, 7839321861, 205683135000, 5312031978672, 135495143785470, 3421536337406913, 85686871818240000, 2130987634616000199, 52682956706683197258, 1295799745309605101520, 31730077997731715070000

Offset: 1

Author

N. J. A. Sloane, May 27 2012

Keywords

• nonn
• easy

Comments

A linear divisibility sequence of order 10. - Peter Bala, May 04 2014

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..200
• Germain Kreweras, Complexite et circuits Euleriens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (58,-1131,8700,-29493,43734,-29493,8700,-1131,58,-1).

Crossrefs

Cf. A212796, A054493.

Programs

Maple

seq(simplify(n/3*(-2 + ( (5 + sqrt(21))/2 )^n + ( (5 - sqrt(21))/2 )^n)^2), n = 1..14); # Peter Bala, May 04 2014

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 A212798(n):
    if n == 1: return 3
    if n == 2: return 294
    universe = make_CnXCk(n, 3)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A212798(n) for n in range(1, 30)])  # Seiichi Manyama, Nov 22 2020

Formula

From Peter Bala, May 04 2014: (Start)

a(n) = n/3*(-2 + ( (5 + sqrt(21))/2 )^n + ( (5 - sqrt(21))/2 )^n)^2 = 3*n*A054493(n-1)^2.

O.g.f.: 3*(x^8 + 40*x^7 - 665*x^6 - 866*x^5 + 5626*x^4 - 866*x^3 - 665*x^2 + 40*x + 1)/( (x - 1)^2*(x^2 - 5*x + 1)^2*(x^2 - 23*x + 1)^2 ). (End)

Extensions

More terms from Peter Bala, May 04 2014