A110041 -id:A110041 - cp's OEIS frontend

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.

1, 0, 0, 1, 10, 112, 1760, 35150, 848932, 24243520, 805036704, 30649435140, 1322299270600, 64008728200384, 3447361661136640, 205070807479444088, 13388424264027157520, 953966524932871436800, 73817914562041635228928

Offset: 0

Marni Mishna, Jul 08 2005

P-recursive.
Starting at n=3, number of symmetric binary matrices with all row sums 3. - R. H. Hardin, Jun 12 2008
From R. J. Mathar, Apr 07 2017: (Start)
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:
0:  1
1:  0 0
2:  0 0 0
3:  0 0 0 1
4:  1 0 6 0 3
5:  0 30 0 70 0 12
6:  70 0 810 0 810 0 70
7:  0 5670 0 19355 0 9660 0 465
This has A001205 on the diagonal. (End)
The traceless (2n) X (2n) binary matrices in that triangle seem to be counted in A002829. - Alois P. Heinz, Apr 07 2017

			(Graphs listed by edgeset)
a(3)=1: {(1,2), (2,3), (3,1)}
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)},

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..320
I. P. Goulden and D. M. Jackson, Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093). [Gives e.g.f.]

Cf. A002829, A110039, A110041.

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

RecurrenceTable[{-b[n] - b[1 + n] + (-2 + 3*n)*b[2 + n] - 14*b[3 + n] + (105 + 30*n)*b[4 + n] + (-69 - 12*n)*b[5 + n] + (582 + 147*n + 9*n^2)* b[6 + n] + (-20 - 6*n)*b[7 + n] + (1160 + 363*n + 27*n^2)*b[8 + n] + (1554 + 255*n + 9*n^2)* b[9 + n] + (-2340 - 414*n - 18*n^2)*b[10 + n] + (-528 - 48*n)*b[11 + n] + (288 + 24*n)*b[12 + n] == 0, b[0] == 1, b[1] == 0, b[2] == 0, b[3] == 1/6, b[4] == 5/12, b[5] == 14/15, b[6] == 22/9, b[7] == 3515/504, b[8] == 30319/1440, b[9] == 10823/162, b[10] == 8385799/37800, b[11] == 510823919/665280}, b, {n, 0, 25}] * Range[0, 25]! (* Vaclav Kotesovec, Oct 23 2023 *)

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).
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)}.
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
Recurrence (of order 8): 12*(27*n^4 - 423*n^3 + 2427*n^2 - 5639*n + 4384)*a(n) = 6*(n-1)*(81*n^4 - 1242*n^3 + 7011*n^2 - 15528*n + 10352)*a(n-1) + 3*(n-2)*(n-1)*(81*n^5 - 1269*n^4 + 7551*n^3 - 20841*n^2 + 29934*n - 16040)*a(n-2) - 3*(n-2)*(n-1)*(135*n^5 - 2115*n^4 + 13287*n^3 - 37537*n^2 + 46430*n - 21848)*a(n-3) + (n-3)*(n-2)*(n-1)*(567*n^5 - 9396*n^4 + 59895*n^3 - 169590*n^2 + 191744*n - 57040)*a(n-4) - 2*(n-4)*(n-3)*(n-2)*(n-1)*(135*n^4 - 1386*n^3 + 5034*n^2 - 6529*n + 648)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^5 - 1566*n^4 + 11367*n^3 - 37080*n^2 + 47872*n - 17424)*a(n-6) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(27*n^4 - 315*n^3 + 1113*n^2 - 1433*n + 348)*a(n-7) - (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(27*n^4 - 315*n^3 + 1320*n^2 - 1946*n + 776)*a(n-8). - Vaclav Kotesovec, Oct 23 2023
a(n) ~ 3^(n/2) * n^(3*n/2) / (2^(n + 1/2) * exp(3*n/2 - sqrt(3*n) + 13/4)) * (1 + 119/(24*sqrt(3*n)) - 2519/(3456*n)). - Vaclav Kotesovec, Oct 27 2023, extended Oct 28 2023

Edited and extended by Max Alekseyev, May 08 2010

A136282 Number of graphs on n labeled nodes with degree at most 3.

Original entry on oeis.org

1, 2, 8, 64, 768, 12068, 236926, 5651384, 160054952, 5284391984, 200375581984, 8620342917808, 416471882713712, 22400989824444576, 1331457489258580672, 86887134810544955072, 6189888588922841477824, 478992737680928902742656, 40082045451011806706919808, 3612470757307682016196841216, 349398857659776033845292636416

Offset: 1

Don Knuth, Mar 31 2008

nonn

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000085 (degree at most 1), A136281-A136286.

Binomial transform of A110041. - Vladeta Jovovic, May 20 2008
Recurrence: 12*(81*n^4 - 837*n^3 + 3375*n^2 - 6171*n + 4192)*a(n) = 6*(243*n^5 - 2511*n^4 + 10665*n^3 - 21969*n^2 + 19476*n - 4624)*a(n-1) + 3*(n-1)*(243*n^6 - 2997*n^5 + 15147*n^4 - 39843*n^3 + 57594*n^2 - 41832*n + 10888)*a(n-2) - 3*(n-2)*(n-1)*(405*n^5 - 3699*n^4 + 13527*n^3 - 22629*n^2 + 14048*n + 388)*a(n-3) + (n-3)*(n-2)*(n-1)*(243*n^5 - 1944*n^4 + 6777*n^3 - 9738*n^2 - 2370*n + 10732)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 999*n^3 + 4968*n^2 - 8646*n + 4906)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^5 - 2916*n^4 + 12933*n^3 - 27990*n^2 + 27978*n - 8948)*a(n-6) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 513*n^3 + 891*n^2 - 357*n - 242)*a(n-7) - (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 513*n^3 + 1350*n^2 - 1608*n + 640)*a(n-8). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 3^(n/2) * exp(sqrt(3*n) - 3*n/2 - 5/4) * n^(3*n/2) / 2^(n + 1/2) * (1 + 71/(24*sqrt(3*n))). - Vaclav Kotesovec, Nov 05 2023
a(n) / A110041(n) ~ 1 + 2/sqrt(3*n). - Vaclav Kotesovec, Nov 06 2023

More terms from Vladeta Jovovic, May 20 2008

cp's OEIS Frontend

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.

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