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A338832 Number of spanning trees in the k_1 X ... X k_j grid graph, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.

%I A338832 #17 Feb 16 2025 08:34:00
%S A338832 1,1,1,4,1,15,1,384,192,56,1,31500,1,209,2415,42467328,1,49766400,1,
%T A338832 2558976,30305,780,1,3500658000000,100352,2911,8193540096000,
%U A338832 207746836,1,76752081000,1,20776019874734407680,380160,10864,4140081,242716067758080000000,1
%N A338832 Number of spanning trees in the k_1 X ... X k_j grid graph, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.
%C A338832 a(n) > 1 precisely when n is composite.
%H A338832 Pontus von Brömssen, <a href="/A338832/b338832.txt">Table of n, a(n) for n = 1..1151</a>
%H A338832 Germain Kreweras, <a href="http://dx.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.
%H A338832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H A338832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%H A338832 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A338832 a(n) = Product_{n_1=0..k_1-1, ..., n_j=0..k_j-1; not all n_i=0} Sum_{i=1..j} (2*(1 - cos(n_i*Pi/k_i))) / Product_{i=1..j} k_i, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.
%e A338832 The partition (2, 2, 1) has Heinz number 18 and the 3 X 3 X 2 grid graph has a(18) = 49766400 spanning trees.
%Y A338832 2 X n grid: A001353(n) = a(2*prime(n-1))
%Y A338832 3 X n grid: A006238(n) = a(3*prime(n-1))
%Y A338832 4 X n grid: A003696(n) = a(5*prime(n-1))
%Y A338832 5 X n grid: A003779(n) = a(7*prime(n-1))
%Y A338832 6 X n grid: A139400(n) = a(11*prime(n-1))
%Y A338832 7 X n grid: A334002(n) = a(13*prime(n-1))
%Y A338832 8 X n grid: A334003(n) = a(17*prime(n-1))
%Y A338832 9 X n grid: A334004(n) = a(19*prime(n-1))
%Y A338832 10 X n grid: A334005(n) = a(23*prime(n-1))
%Y A338832 n X n grid: A007341(n) = a(prime(n-1)^2)
%Y A338832 m X n grid: A116469(m,n) = a(prime(m-1)*prime(n-1))
%Y A338832 2 X 2 X n grid: A003753(n) = a(4*prime(n-1))
%Y A338832 2 X n X n grid: A067518(n) = a(2*prime(n-1)^2)
%Y A338832 n X n X n grid: A071763(n) = a(prime(n-1)^3)
%Y A338832 2 X ... X 2 grid: A006237(n) = a(2^n)
%K A338832 nonn
%O A338832 1,4
%A A338832 _Pontus von Brömssen_, Nov 11 2020