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A000986
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Number of n X n symmetric matrices with (0,1) entries and all row sums 2.
(Formerly M3548 N1437)
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11
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1, 0, 1, 4, 18, 112, 820, 6912, 66178, 708256, 8372754, 108306280, 1521077404, 23041655136, 374385141832, 6493515450688, 119724090206940, 2337913445039488, 48195668439235612, 1045828865817825264, 23826258064972682776, 568556266922455167040
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OFFSET
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0,4
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COMMENTS
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a(n) is the number of simple labeled graphs on n nodes with all vertices of degree 1 or 2.
These are the row sums of the following triangle which shows the number of symmetric n X n {0,1} matrices with row and column sums 2 refined for trace t, 0 <= t <= n:
0: 1
1: 0 0
2: 0 0 1
3: 1 0 3 0
4: 3 0 12 0 3
5: 12 0 70 0 30 0
6: 70 0 465 0 270 0 15
7: 465 0 3507 0 2625 0 315 0
See also A001205 for column t=0. (End)
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.8.
Herbert S. Wilf, Generatingfunctionology, p. 104.
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LINKS
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FORMULA
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E.g.f.: (1-x)^(-1/2)*exp(-x-x^2/4 + x/((2*(1-x)))).
Sum_{a_1=0..n} Sum_{c=0..min(a_1, n - a_1)} Sum_{b=0..floor((n - a_1 - c)/2)} (
(-1)^((n - a_1 - 2b - c) + b) n!(2a_{1})!}{% 2^{n+a_{1}-2c}a_{1}!(n-a_{1}-2b-c)!b!(2c)!(a_{1}-c)!}$
Sum_{a_1=0..n} Sum_{c=0..min(a_1, n - a_1)} Sum_{b=0..floor((n - a_1 - c)/2)} ((-1)^((n - a_1 - 2b - c) + b)*n!*(2a_1)!) / (2^(n + a_1 - 2c)*a_1!*(n - a_1 - 2b - c)!*b!*(2c)!*(a_1 - c)!). - Shanzhen Gao, Jun 05 2009
Conjecture: 2*a(n) +2*(-2*n+1)*a(n-1) +2*(n^2-2*n-1)*a(n-2) -2*(n-2)*(n-4)*a(n-3) +(n-1)*(n-2)*(n-3)*a(n-4) -(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 04 2013
Recurrence: 2*a(n) = 4*(n-1)*a(n-1) - 2*(n-3)*(n-1)*a(n-2) - (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ n^n * exp(sqrt(2*n)-n-3/2) / sqrt(2) * (1 + 43/(24*sqrt(2*n))). - Vaclav Kotesovec, Feb 13 2014
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MAPLE
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a:= proc(n) option remember;
`if`(n<2, 1-n, add(binomial (n-1, k-1)
*(k! +`if`(k>2, (k-1)!, 0))/2 *a(n-k), k=2..n))
end:
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MATHEMATICA
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a=1/(2(1-x))-1/2-x/2; b=(Log[1/(1-x)]-x-x^2/2)/2;
Range[0, 20]! CoefficientList[Series[Exp[a + b], {x, 0, 20}], x]
(* Second program: *)
a[n_] := a[n] = If[n<2, 1-n, Sum[Binomial[n-1, k-1]*(k! + If[k>2, (k-1)!, 0])/2*a[n-k], {k, 2, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 20 2017, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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