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A320653
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a(n) is the associated coefficient of the n-uniform simplex.
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0
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2, 21, 588, 28230, 2092206, 220611384, 31373370936, 5785037767440, 1342136211324090, 382559909729171328, 131411551493995125828, 53537846795391076075776, 25523603120175022166538150, 14076445847378724286239575040, 8892219411843450738850246324464
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OFFSET
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2,1
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COMMENTS
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The associated coefficient of a hypergraph is the weight assigned to it in the generalized Harary-Sachs formula (arxiv link to be posted). For example, the 2-uniform simplex is a triangle and a(2) = 2. Famously the codegree 3 coefficient of the adjacency characteristic polynomial of a graph is -a(2)(# of triangles in G). The quantity a(n)=C_n as mentioned in Cooper and Dutle wherein the authors computed the values up to n=5.
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LINKS
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FORMULA
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Let P(n,2) denote the set of partitions of n where each part is of size at least 2. Let L(p) denote the length of p, let p(i) denote the size of part i of p, and let V(p,i) denote the number of parts of p which have size i. For p in P(n,2) let f(n,p)= n!/((Product_{i=1..L(p)} p(i))(Product_{i=2..n}V(p,i)). Then a(n) = (1/((n-1)*(n+1)^2))*Sum_{p in P(n+1,2)}(f(n+1,p)*Product_{i=1 .. L(p)}(n^(p(i)) + (-1)^(p(i)+1))).
a(n) = exp(n*log(n)(2+o(1)).
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PROG
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(Sage)
def simplex_coefficient(n):
P=Partitions(n+1, min_part=2)
x=0
for p in P:
E = p.evaluation()
tau=1
c = 1
d=1
for i in p:
tau = tau*(n^i+(-1)^(i+1))
c = c * i
for i in E:
d = d * factorial(i)
x= x + tau/(c * d)
return factorial(n+1)*x/((n-1)*(n+1)^2)
[simplex_coefficient(n) for n in range(2, 4)]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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