cs=Last /@ {c[0]->-(11/15360),c[1]->-(19/46080),c[2]->-(187/138240),c[3]->-(169/138240),c[4]->-(169/138240),c[5]->-(187/138240),c[6]->-(19/46080),c[7]->-(11/15360)}

-138240*cs

pol=-(1/4) Diag[1,ar[1,2]]+(1/4) Diag[1,ar[2,1]]+(1/32) Diag[h[1],ar[1,2]]+(5/96) Diag[h[1],ar[2,1]]-((19 Diag[((h[1])^3),ar[1,2]])/46080)-((11 Diag[((h[1])^3),ar[2,1]])/15360)+((37 Diag[((h[1])^5),ar[1,2]])/4644864)+((13 Diag[((h[1])^5),ar[2,1]])/1161216)+(5/96) Diag[h[2],ar[1,2]]+(1/32) Diag[h[2],ar[2,1]]-((169 Diag[((h[1])^2) h[2],ar[1,2]])/138240)-((187 Diag[((h[1])^2) h[2],ar[2,1]])/138240)+((1279 Diag[((h[1])^4) h[2],ar[1,2]])/34836480)+((2669 Diag[((h[1])^4) h[2],ar[2,1]])/69672960)-((187 Diag[h[1] ((h[2])^2),ar[1,2]])/138240)-((169 Diag[h[1] ((h[2])^2),ar[2,1]])/138240)+((125 Diag[((h[1])^3) ((h[2])^2),ar[1,2]])/1741824)+((2539 Diag[((h[1])^3) ((h[2])^2),ar[2,1]])/34836480)-((11 Diag[((h[2])^3),ar[1,2]])/15360)-((19 Diag[((h[2])^3),ar[2,1]])/46080)+((2539 Diag[((h[1])^2) ((h[2])^3),ar[1,2]])/34836480)+((125 Diag[((h[1])^2) ((h[2])^3),ar[2,1]])/1741824)+((2669 Diag[h[1] ((h[2])^4),ar[1,2]])/69672960)+((1279 Diag[h[1] ((h[2])^4),ar[2,1]])/34836480)+((13 Diag[((h[2])^5),ar[1,2]])/1161216)+((37 Diag[((h[2])^5),ar[2,1]])/4644864) /. {
Diag[_, ar[2,1]] -> 0,
Diag[hs_, ar[1,2]] :> hs
}

cs={19/46080, 169/138240, 187/138240, 11/15360}

138240cs

GCD@@(138240cs)

Expand[4pol]

cs1=Table[Coefficient[4*17418240*pol, h[1]^(5-k)h[2]^k], {k, 0, 5}]

ser=1+(-(x/8)-((5 y)/24)) h+(((57 (x^3)+169 (x^2) y+187 x (y^2)+99 (y^3)) (h^3))/34560)+(1/17418240)(-555 (x^5)-2558 (x^4) y-5000 (x^3) (y^2)-5078 (x^2) (y^3)-2669 x (y^4)-780 (y^5)) (h^5)+(1/50164531200)(31989 (x^7)+205357 (x^6) y+592097 (x^5) (y^2)+964985 (x^4) (y^3)+949967 (x^3) (y^4)+565943 (x^2) (y^5)+186139 x (y^6)+34275 (y^7)) (h^7)+(1/397303087104000)(-5104911 (x^9)-42750073 (x^8) y-165124408 (x^7) (y^2)-376113312 (x^6) (y^3)-552802446 (x^5) (y^4)-542218746 (x^4) (y^5)-352835712 (x^3) (y^6)-146191048 (x^2) (y^7)-33851563 x (y^8)-4104741 (y^9)) (h^9)+((O[h])^10)

FindSequenceFunction[{24,34560,17418240}]

ser /. {x|y -> 1}

FindSequenceFunction[{1/3, -2/135, 13/13608, -431/6123600, 27113/4849891200}]

ser /. {x->1, y->-1}

FindSequenceFunction[{12,1440,90720, 4838400, 239500800}]

Google finds 12, 1440, 90720, 4838400, 239500800 !

{24/12,34560/1440,17418240/90720, 50164531200/4838400, 397303087104000/239500800}

FactorInteger[{2,24,192,10368,1658880}]

FactorInteger[{2,24,192,10368,1658880}/Range[5]!]

Simplify[ser /. {x->(a+b)/2, y->(a-b)/2}]