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A054559
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Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.
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7
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30, 180, 630, 1680, 3780, 7560, 13860, 23760, 38610, 60060, 90090, 131040, 185640, 257040, 348840, 465120, 610470, 790020, 1009470, 1275120, 1593900, 1973400, 2421900, 2948400, 3562650, 4275180, 5097330, 6041280, 7120080, 8347680, 9738960, 11309760, 13076910
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OFFSET
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5,1
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COMMENTS
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Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=8.
Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row 3 of B equals -a(n+2). - T. D. Noe, May 01 2011
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REFERENCES
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V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.
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LINKS
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FORMULA
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a(n) = 30*C(n,5) = 30*A000389(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/4.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Vincenzo Librandi, Apr 29 2012
Sum_{n>=5} 1/a(n) = 1/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = 8*log(2)/3 - 131/72. (End)
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MATHEMATICA
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CoefficientList[Series[30/(1-x)^6, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 29 2012 *)
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PROG
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(Magma) I:=[30, 180, 630, 1680, 3780, 7560]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Apr 29 2012
(PARI) x='x+O('x^30); Vec(serlaplace(x^5*exp(x)/4)) \\ G. C. Greubel, Nov 23 2017
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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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