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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*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)}
$\sum_{a_2=0}^{n} \sum_{d_2=0}^{\min \{\lfloor (3n-2a_2)/2\rfloor, \lfloor n/2\rfloor ,(n-a_2)\}} \sum_{d_{3}=0}^{\min \{\lfloor (3n-2a_2-2d_2)/3\rfloor, \lfloor (n-2d_2)/3\rfloor, (n-a_2-d_2)\}} \sum_{d_1=0}^{\min \{(3n-2a_2-2d_2-3d_3), (n-2d_2-3d_3)\}} \sum_{b=0}^{\min \{\lfloor (3n-2a_2-2d_2-3d_3-d_1)/4\rfloor, \lfloor (n-d_2-d_3-a_2)/2\rfloor\}} \sum_{c=0}^{\min \{\lfloor (3n-2a_2-2d_2-3d_3-d_1-4b)/6\rfloor, \lfloor (n-a_2-2b-d_2-d_3)/2\rfloor \}} \sum_{a_1=\lceil (3n-(2a_2+4b+6c+d_1+2d_2+3d_3))/2 \rceil}^{\lfloor (3n-(2a_2+4b+6c+d_1+2d_2+3d_3))/2 \rfloor} \frac{(-1)^{a_2+b+d_2}n!(2a_1+d_1)!}{2^{n+a_1-c-d_3} 3^{n-a_2-2b-d_2-c}a_1!a_2!b!c!d_1!d_2!d_3!(n-a_2-2b-d_2-2c-d_3)!}$ [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]
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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)},
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