Pensieve header: Solution of HW5 Q1.

Question 1.

m is the number of vertices, l the number of edges.

Module[{l,m},Log[Sum[l=3m/2;  (((3m)!(ϵ^m))/((2^l)l!(6^m)m!)), {m,0,30,2}]+((O[ϵ])^31)]]

Sum[l=2m;  (((4m)!(ϵ^m))/((2^l)l!(24^m)m!)), {m,0,15}]+((O[ϵ])^16)

Module[{l,m},Log[Sum[l=2m;  (((4m)!(ϵ^m))/((2^l)l!(24^m)m!)), {m,0,15}]+((O[ϵ])^16)]]

(1/(2^3))

(1/(2 4!))+(1/(2^4))

See also https://oeis.org/A226259

Question 2.

ℊ((Subscript[m, k]^ij))=Exp[((Subscript[α, i])+(Subscript[α, j]))(Subscript[a, k])+((e^(-(Subscript[α, j])))(Subscript[ξ, i])+(Subscript[ξ, j]))(Subscript[x, k])].