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A110040 Number of {2,3}-regular graphs, i.e., labeled simple graphs (no multi-edges or loops) on n vertices, each of degree 2 or 3. 4

%I

%S 1,0,0,1,10,112,1760,35150,848932,24243520,805036704,30649435140,

%T 1322299270600,64008728200384,3447361661136640,205070807479444088,

%U 13388424264027157520,953966524932871436800,73817914562041635228928

%N Number of {2,3}-regular graphs, i.e., labeled simple graphs (no multi-edges or loops) on n vertices, each of degree 2 or 3.

%C P-recursive.

%C Starting at n=3, number of symmetric binary matrices with all row sums 3. - _R. H. Hardin_, Jun 12 2008

%C From _R. J. Mathar_, Apr 07 2017: (Start)

%C These are the row sums of the following matrix, which counts symmetric n X n {0,1} matrices with each row and column sum equal to 3 and trace t, 0 <= t <= n:

%C 0: 1

%C 1: 0 0

%C 2: 0 0 0

%C 3: 0 0 0 1

%C 4: 1 0 6 0 3

%C 5: 0 30 0 70 0 12

%C 6: 70 0 810 0 810 0 70

%C 7: 0 5670 0 19355 0 9660 0 465

%C This has A001205 on the diagonal. (End)

%C The traceless (2n) X (2n) binary matrices in that triangle seem to be counted in A002829. - _Alois P. Heinz_, Apr 07, 2017

%D Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From _Shanzhen Gao_, Jun 05 2009]

%H I. P. Goulden and D. M. Jackson, <a href="http://dx.doi.org/10.1137/0607007">Labelled graphs with small vertex degrees and P-recursiveness</a>, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093). [Gives e.g.f.]

%F Satisfies the linear recurrence: (-150917976*n^2 - 105258076*n^3 - 1925*n^9 - 13339535*n^5 - 45995730*n^4 - 357423*n^7 - 2637558*n^6 - 120543840*n - n^11 - 66*n^10 - 39916800 - 32670*n^8)*a(n) + (-11028590*n^4 - 65*n^9 - n^10 - 2310945*n^5 - 1860*n^8 - 30810*n^7 - 326613*n^6 - 80627040*n - 39916800 - 34967140*n^3 - 70290936*n^2)*a(n + 1) + (3*n^10 - 39916800 + 187*n^9 + 5076*n^8 + 78558*n^7 + 761103*n^6 + 4757403*n^5 + 18949074*n^4 + 44946092*n^3 + 51046344*n^2 - 793440*n)*a(n + 2) + (-93139200 - 16175880*n^3 - 56394184*n^2 - 110513760*n - 2854446*n^4 - 14*n^8 - 840*n^7 - 21756*n^6 - 317520*n^5)*a(n + 3) + (45780*n^6 + 1785*n^7 + 111580320*n^2 + 660450*n^5 + 5856270*n^4 + 32645865*n^3 + 174636000 + 213450300*n + 30*n^8)*a(n + 4) + (-22952160 - 681*n^6 - 16419*n^5 - 217995*n^4 - 8082204*n^2 - 20896956*n - 12*n^7 - 1721253*n^3)*a(n + 5) + (1804641*n^3 + 9*n^7 + 14442*n^5 + 208920*n^4 + 32266080 + 9307488*n^2 + 26537388*n + 552*n^6)*a(n + 6) + (-158400 - 15160*n - 3994*n^3 - 31072*n^2 - 6*n^5 - 248*n^4)*a(n + 7) + (20123*n^3 + 706210*n + 27*n^5 + 170067*n^2 + 1148400 + 1173*n^4)*a(n + 8) + (7899*n^2 + 60684*n + 444*n^3 + 9*n^4 + 170940)*a(n + 9) + (-6894*n - 25740 - 18*n^3 - 612*n^2)*a(n + 10) + (-48*n - 528)*a(n + 11) + 24*a(n + 12).

%F Differential equation satisfied by the exponential generating function {F(0) = 1, 9*t^4*(t^4 + t - 2 + 3*t^2)^2*(d^2/dt^2)F(t) + 3*t*(t^4 + t - 2 + 3*t^2)*(10*t^8 + 34*t^3 - 16*t + 16*t^6 - 2*t^5 - 24*t^2 - 4*t^7 + 8 + t^10 - 14*t^4)*(d/dt)F(t) - t^3*(-22*t^2 + t^8 - 24*t^3 + t^9 + 8*t^7 + 14*t^6 + 15*t^5 + 12 + 16*t + 9*t^4)*(t^4 + t - 2 + 3*t^2)*F(t)}.

%F Sum_{a_2 = 0..n} Sum_{d_2 = 0..min(floor((3n - 2a_2)/2), floor(n/2), n - a_2)} Sum_{d_3 = 0..min(floor((3n - 2a_2 - 2d_2)/3), floor((n-2d_2)/3), n - a_2 - d_2} Sum_{d_1 = 0..min(3n - 2a_2 - 2d_2 - 3d_3, n - 2d_2 - 3d_3) Sum_{b = 0..min(floor((3n - 2a_2 - 2d_2 - 3d_3 - d_1)/4), floor((n - d_2 - d_3 - a_2)/2)} Sum_{c = 0..min(floor((3n - 2a_2 - 2d_2 - 3d_3 - d_1 - 4b)/6), floor((n - a_2 - 2b - d_2 - d_3)/2))} Sum_{a_1 = ceiling((3n - (2a_2 + 4b + 6c + d_1 + 2d_2 + 3d_3))/2)..floor((3n - (2a_2 + 4b + 6c + d_1 + 2d_2 + 3d_3))/2)} (-1)^(a_2 + b + d_2)*n!*(2a_1 + d_1)!/(2^(n + a_1 - c - d_3)*3^(n - a_2 - 2b - d_2 - c)*a_1!*a_2!*b!*c!*d_1!*d_2!*d_3!*(n - a_2 - 2b - d_2 - 2c - d_3)!). - _Shanzhen Gao_, Jun 05 2009

%e (Graphs listed by edgeset)

%e a(3)=1: {(1,2), (2,3), (3,1)}

%e a(4)=10: {(1,2), (2,3), (3,4), (4,1)}, {(1,2), (2,3), (3,4), (4,1), (1,4)}, {(1,2), (2,3), (3,4), (4,1), (2,3)}, {(1,2), (2,4), (3,4), (1,3)}, {(1,2), (2,4), (3,4), (1,3), (2,3)}, {(1,2), (2,4), (3,4), (1,3), (1,4)}, {(1,3), (2,3), (2,4), (1,4)}, {(1,3), (2,3), (2,4), (1,4), (1,2)}, {(1,3), (2,3), (2,4), (1,4), (3,4)}, {(1,2), (1,3), (1,4) (2,3), (2,4), (3,4)},

%Y Cf. A002829, A110039, A110041.

%Y Cf. A000986 (sums 2), A000085 (sums 1), A139670 (sums 3).

%K easy,nonn

%O 0,5

%A _Marni Mishna_, Jul 08 2005

%E Edited and extended by _Max Alekseyev_, May 08 2010

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Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)