More information about A099398 and A099399.

The rationals A(n) = A099398(n)/A099399(n) := Catalan(n)/((n+2)*2^(2*n-1))= 8*(2*n-1)!!/(2*(n+2))!!  

with the Catalan numbers A000108(n) and the double factorials (2*n-1)!!:=A001147(n) (with (-1)!!:=1) and 

(2*n)!!:=A000165(n) are, for n=0,...,30:

[1, 1/6, 1/16, 1/32, 7/384, 3/256, 33/4096, 143/24576, 143/32768, 221/65536, 4199/1572864, 2261/1048576, 

7429/4194304, 37145/25165824, 334305/268435456, 570285/536870912, 1964315/2147483648, 3411705/4294967296, 

23881935/34359738368, 42077695/68719476736, 149184555/274877906944, 265937685/549755813888, 

3811773485/8796093022208, 6861192273/17592186044416, 24805848987/70368744177664, 135054066707/422212465065984, 

327988447717/1125899906842624, 599427163069/2251799813685248, 6593698793759/27021597764222976, 

4041299260691/18014398509481984, 238436656380769/1152921504606846976]

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In the model with zeroth order potential V0(phi):= ((m^2)/2)*phi^2 + (lambda/4!)*phi^4 one has for the one-loop 

effective potential a temperature T dependent piece V1T(y) of the form

(2/((pi^2)*(kT)^4)* V1T(y) = - 1/45 + (1/3)*y  - (4/3)*y^(3/2) - (1/2)*y^2*ln(y/c) + sum(((-1)^(k+1))*Zeta(2*k+1)*A(k)*y^(k+2),k=1..infty)  
 
with  y:=(m^2(phi))/(2*pi*k*T)^2 with Boltzmann's constant k,  m^2(phi):= m^2 + (lambda/2) phi^2 and the 

constant c:=4*exp(3/2-2*gamma)=5.65117241... gamma= .577215664... is the Euler-Mascheroni constant and Zeta(n) is Euler's 

(also Riemann's) Zeta-function. 

Compare with the appendix C, p.3340, of the Dolan-Jackiw ref. given in A099398 

and the integral h_1(y) of the appendix A of the Kapusta ref. given in A099398.

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The above sum in (2/((pi^2)*(kT)^4)* V1T(y) is, for k=1..20 (instead of infty)  (Maple9 with 10-digit floating-point arithmetic) 

.2003428172*y^3 - .06480798468*y^4 + .03151091491*y^5 - .01826577800*y^6 + .01172454128*y^7 - .008057629280*y^8 + 

+ .005818862876*y^9 - .004364047000*y^10 + .003372198817*y^11 - .002669653577*y^12 + .002156257886*y^13 - 

- .001771211677*y^14 + .001476009697*y^15 - .001245383176*y^16 + .001062238589*y^17 - .0009147054516*y^18 + 

+ .0007943494712*y^19 - .0006950557872*y^20 + .0006123110508*y^21 - .0005427302496 *y^22 .

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