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A344638
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Number of compositions of graph K_4 X P_n.
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0
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15, 1548, 168386, 18328142, 1994963186, 217145777610, 23635668646510, 2572671863723654, 280027640317060130, 30480171391948784938, 3317675523140039250350, 361119061152982241895174, 39306730094143339494849314, 4278420047285488959291378858, 465693230069569504343096792622
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = 112*a(n-1) - 346*a(n-2) + 306*a(n-3) - 57*a(n-4) + 2*a(n-5) for n >= 6.
G.f.: (-15 + 132*x - 200*x^2 + 72*x^3 - 5*x^4)/(-1 + 112*x - 346*x^2 + 306*x^3 - 57*x^4 + 2*x^5).
For n>1, a(n) = z * M^(n-1) * z^T, where z is the 1 X 15 row vector [1,1,1,...,1], z^T is its transpose (a 15 X 1 column vector of 1's), and M is the 15 X 15 matrix
[[16, 12, 12, 12, 12, 12, 12, 9, 9, 9, 8, 8, 8, 8, 5],
[12, 8, 10, 10, 9, 10, 10, 6, 8, 8, 6, 6, 7, 7, 4],
[12, 10, 8, 9, 10, 10, 10, 8, 6, 8, 6, 7, 6, 7, 4],
[12, 10, 9, 8, 10, 10, 10, 8, 6, 8, 7, 6, 7, 6, 4],
[12, 9, 10, 10, 8, 10, 10, 6, 8, 8, 7, 7, 6, 6, 4],
[12, 10, 10, 10, 10, 8, 9, 8, 8, 6, 7, 6, 6, 7, 4],
[12, 10, 10, 10, 10, 9, 8, 8, 8, 6, 6, 7, 7, 6, 4],
[ 9, 6, 8, 8, 6, 8, 8, 4, 7, 7, 5, 5, 5, 5, 3],
[ 9, 8, 6, 6, 8, 8, 8, 7, 4, 7, 5, 5, 5, 5, 3],
[ 9, 8, 8, 8, 8, 6, 6, 7, 7, 4, 5, 5, 5, 5, 3],
[ 8, 6, 6, 7, 7, 7, 6, 5, 5, 5, 4, 5, 5, 5, 3],
[ 8, 6, 7, 6, 7, 6, 7, 5, 5, 5, 5, 4, 5, 5, 3],
[ 8, 7, 6, 7, 6, 6, 7, 5, 5, 5, 5, 5, 4, 5, 3],
[ 8, 7, 7, 6, 6, 7, 6, 5, 5, 5, 5, 5, 5, 4, 3],
[ 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2]].
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EXAMPLE
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Here are the a(1) = 15 compositions of the graph K_4 x P_1 = K_4, where the first block represents all four vertices of K_4 in the same partition (called "a"), the second block shows three vertices in partition "a" and the fourth vertex in its own partition (called "b"), and so on, up to the last block which shows all four vertices each in its own partition:
aa aa aa ba ab bb ab ab aa ba cb ac ab ba ab
aa ab ba aa aa aa ab ba bc ca aa ab ca ac cd
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MATHEMATICA
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M = {{16, 12, 12, 12, 12, 12, 12, 9, 9, 9, 8, 8, 8, 8, 5},
{12, 8, 10, 10, 9, 10, 10, 6, 8, 8, 6, 6, 7, 7, 4},
{12, 10, 8, 9, 10, 10, 10, 8, 6, 8, 6, 7, 6, 7, 4},
{12, 10, 9, 8, 10, 10, 10, 8, 6, 8, 7, 6, 7, 6, 4},
{12, 9, 10, 10, 8, 10, 10, 6, 8, 8, 7, 7, 6, 6, 4},
{12, 10, 10, 10, 10, 8, 9, 8, 8, 6, 7, 6, 6, 7, 4},
{12, 10, 10, 10, 10, 9, 8, 8, 8, 6, 6, 7, 7, 6, 4},
{9, 6, 8, 8, 6, 8, 8, 4, 7, 7, 5, 5, 5, 5, 3},
{9, 8, 6, 6, 8, 8, 8, 7, 4, 7, 5, 5, 5, 5, 3},
{9, 8, 8, 8, 8, 6, 6, 7, 7, 4, 5, 5, 5, 5, 3},
{8, 6, 6, 7, 7, 7, 6, 5, 5, 5, 4, 5, 5, 5, 3},
{8, 6, 7, 6, 7, 6, 7, 5, 5, 5, 5, 4, 5, 5, 3},
{8, 7, 6, 7, 6, 6, 7, 5, 5, 5, 5, 5, 4, 5, 3},
{8, 7, 7, 6, 6, 7, 6, 5, 5, 5, 5, 5, 5, 4, 3},
{5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2}};
w = Table[1, {15}]; Join[{15}, Table[Transpose[w] . MatrixPower[M, n, w], {n, 1, 40}]]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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