Let's assume that each term T(n) is given by a polynomial formula:
T(n)=a0+a1n+a2n2+a3n3+a4n4 where the ai's are constants and 0≤n≤4 (that is, n is between 0 and 4 inclusive). T(0)=6; T(1)=38; T(2)=932; T(3)=1087; T(4)=12437.
T(0)=a0 so a0=6. If we subtract 6 from the other T values we get the equations:
(1) a1+a2+a3+a4=32
(2) 2a1+4a2+8a3+16a4=926
(3) 3a1+9a2+27a3+81a4=1081
(4) 4a1+16a2+64a3+256a4=12431
(5)=(2)-2(1)=2a2+6a3+14a4=862⇒a2+3a3+7a4=431
(6)=(3)-3(1)=6a2+24a3+78a4=985
(7)=(4)-4(1)=12a2+60a3+252a4=12303⇒4a2+20a3+84a4=4101
(8)=(6)-6(5)=6a3+36a4=-1601
(9)=(7)-4(5)=8a3+56a4=2377
(10)=3(9)-4(8)=24a4=13535, a4=13535/24⇒
a3=-43807/12⇒
a2=178441/24⇒
a1=-51797/12.
T(n)=6-51797n/12+178441n2/24-43807n3/12+13535n4/24.
The next three terms according to this formula would be 60451, 184133, 436022.
However, the coefficients are so ugly that there's probably another, simpler rule which applies.