A339257 Number of spanning trees in the n X 5 king graph.

1, 27648, 146356224, 698512774464, 3271331573452800, 15258885095892902976, 71111090441547013886784, 331335100372867196224868352, 1543757070688065237574186369344, 7192607774929149127350811889484864, 33511424900308657559195109303117533184, 156134620449573478209362729027690283037248

Offset: 1

Author

Seiichi Manyama, Nov 29 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, King Graph
• Eric Weisstein's World of Mathematics, Spanning Tree

Crossrefs

Column 5 of A338029.

Cf. A003779.

Programs

Python

# Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(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))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A338029(n, k):
    if n == 1 or k == 1: return 1
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
def A339257(n):
    return A338029(n, 5)
print([A339257(n) for n in range(1, 15)])

Formula

Empirical g.f.: -x*(218700000000*x^8 - 2040471000000*x^7 + 538526880000*x^6 + 311791396500*x^5 - 17462695797*x^4 - 80280747*x^3 + 10513308*x^2 - 21759*x - 1) / (656100000000*x^8 - 4293081000000*x^7 + 4819127400000*x^6 - 930215250900*x^5 + 51621632181*x^4 - 1033572501*x^3 + 5949540*x^2 - 5889*x + 1). - Vaclav Kotesovec, Dec 09 2020