0,5

P-recursive

Starting at n=3, number of symmetric binary matrices with all row sums 3. - Ron Hardin (rhhardin(AT)att.net), Jun 12 2008

Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]

Recurrence satisfied by the sequence: {a(1) = 0, a(2) = 0, a(10) = 805036704, a(3) = 1, a(4) = 10, a(5) = 112, a(6) = 1760, a(7) = 35150, a(8) = 848932, a(9) = 24243520, a(11) = 30649435140, ( - 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 - 115160*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), a(0) = 1}

Differential equation satisfied by the exponential generating function {F(0) = 1, 9*t^4*(t^4 + t - 2 + 3*t^2)^2*diff(diff(F(t), t), 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)*diff(F(t), 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)}

$\dsum\limits_{a_{2}=0}^{n}\dsum\limits_{d_{2}=0}^{\min \{\lfloor (3n-2a_{2})/2\rfloor ,\lfloor n/2\rfloor ,(n-a_{2})\}}\dsum\limits_{d_{3}=0}^{\min \{\lfloor (3n-2a_{2}-2d_{2})/3\rfloor ,\lfloor (n-2d_{2})/3\rfloor ,(n-a_{2}-d_{2})\}}\dsum\limits_{d_{1}=0}^{\min \{(3n-2a_{2}-2d_{2}-3d_{3}),(n-2d_{2}-3d_{3})\}}\dsum\limits_{b=0}^{\min \{\lfloor (3n-2a_{2}-2d_{2}-3d_{3}-d_{1})/4\rfloor ,\lfloor (n-d_{2}-d_{3}-a_{2})/2\rfloor \}}\dsum\limits_{c=0}^{\min \{\lfloor (3n-2a_{2}-2d_{2}-3d_{3}-d_{1}-4b)/6\rfloor ,\lfloor (n-a_{2}-2b-d_{2}-d_{3})/2\rfloor \}}\dsum\limits_{a_{1}=\lceil (3n-(2a_{2}+4b+6c+d_{1}+2d_{2}+3d_{3}))/2\rceil }^{\lfloor (3n-(2a_{2}+4b+6c+d_{1}+2d_{2}+3d_{3}))/2\rfloor }\frac{% (-1)^{a_{2}+b+d_{2}}n!(2a_{1}+d_{1})!}{% 2^{n+a_{1}-c-d_{3}}3^{n-a_{2}-2b-d_{2}-c}a_{1}!a_{2}!b!c!d_{1}!d_{2}!d_{3}!(n-a_{2}-2b-d_{2}-2c-d_{3})!% }$ [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]

(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)},

Cf. A000986, A110039, A110041, A002829.

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

Sequence in context: A066275 A046164 A014484 this_sequence A129866 A024130 A104520

Adjacent sequences: A110037 A110038 A110039 this_sequence A110041 A110042 A110043

easy,nonn,more

Marni Mishna (marni.mishna(AT)inria.fr), Jul 08 2005