

A099398


Numerators of rationals (in lowest terms) used in a certain high temperature expansion.


4



1, 1, 1, 1, 7, 3, 33, 143, 143, 221, 4199, 2261, 7429, 37145, 334305, 570285, 1964315, 3411705, 23881935, 42077695, 149184555, 265937685, 3811773485, 6861192273, 24805848987, 135054066707, 327988447717, 599427163069, 6593698793759
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OFFSET

0,5


COMMENTS

The rationals A(n) defined below appear in the expansion of the oneloop effective potential V1(y) for the thermal phi^4 model. See the DolanJackiw, Kapusta and Quir/'os references. The expansion variable is y:=(m^2(phi))/(2*pi*k*T)^2 with Boltzmann's constant k, the (absolute) temperature T and m^2(phi):= m^2 + (lambda/2) phi^2 if the coupling constant is lambda/4! and the mass is m.
The relevant expansion of part of the thermal oneloop effective potential is ((pi^2)*((k*T)^4)/2)*sum(A(n)*Zeta(2*n+1)*(1)^(n+1)*y^(n+2),n=1..infty) with Riemann's Zeta function. The expansion parameter y is given above. See the W. Lang link for more details.


REFERENCES

J. I. Kapusta, Finitetemperature field theory, Cambridge University Press, 1989.
M. Quirós, Field theory at finite temperature and phase transitions, Helv. Phys. Acta 67 (1994) 451583.


LINKS

Table of n, a(n) for n=0..28.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys.Rev. D9,12 (1974) 332041.
Wolfdieter Lang, Rationals A(n) and more.


FORMULA

a(n) = numerator(A(n)) with A(n):= Catalan(n)/((n+2)*2^(2*n1)) where Catalan(n):=A000108(n)=binomial(2*n, n)/(n+1).
a(n) = numerator(8*(2*n1)!!/((2*(n+2))!!)) with the double factorials (2*n1)!!:=A001147(n) (with (1)!!:=1) and (2*n)!!:=A000165(n).


EXAMPLE

Rationals A(n):=A099398(n)/A099399(n), n>=0: 1/1, 1/6, 1/16, 1/32, 7/384, ...


CROSSREFS

The denominators are given in A099399.
Sequence in context: A290235 A225825 A199927 * A272276 A271597 A272505
Adjacent sequences: A099395 A099396 A099397 * A099399 A099400 A099401


KEYWORD

nonn,frac,easy


AUTHOR

Wolfdieter Lang, Nov 10 2004


STATUS

approved



