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 A000986 Number of n X n symmetric matrices with (0,1) entries and all row sums 2. (Formerly M3548 N1437) 11
 1, 0, 1, 4, 18, 112, 820, 6912, 66178, 708256, 8372754, 108306280, 1521077404, 23041655136, 374385141832, 6493515450688, 119724090206940, 2337913445039488, 48195668439235612, 1045828865817825264, 23826258064972682776, 568556266922455167040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n) is the number of simple labeled graphs on n nodes with all vertices of degree 1 or 2. From R. J. Mathar, Apr 07 2017: (Start) 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) REFERENCES 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..445 (first 101 terms from T. D. Noe) H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), vol 3, 653-659. H. Gupta, >Enumeration of symmetric matrices (annotated scanned copy) Zhonghua Tan, Shanzhen Gao, H. Niederhausen, Enumeration of (0,1) matrices with constant row and column sums, Appl. Math. - A Journal of chin. Univ. 21 (4) (2006) 479-486. FORMULA 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 MAPLE 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: seq(a(n), n=0..30);  # Alois P. Heinz, Feb 24 2011 MATHEMATICA 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 *) CROSSREFS Cf. A000985, A001205. Sequence in context: A003708 A327679 A330353 * A143920 A233534 A113356 Adjacent sequences:  A000983 A000984 A000985 * A000987 A000988 A000989 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 21 17:26 EDT 2020. Contains 337919 sequences. (Running on oeis4.)