OFFSET
0,5
COMMENTS
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
REFERENCES
Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From Shanzhen Gao, Jun 05 2009]
LINKS
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.]
FORMULA
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
EXAMPLE
(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)},
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Marni Mishna, Jul 08 2005
EXTENSIONS
Edited and extended by Max Alekseyev, May 08 2010
STATUS
approved