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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 (list; graph; refs; listen; history; text; internal format)



The rationals A(n) defined below appear in the expansion of the one-loop effective potential V1(y) for the thermal phi^4 model. See the Dolan-Jackiw, 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 one-loop 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.


J. I. Kapusta, Finite-temperature field theory, Cambridge University Press, 1989.

M. Quirós, Field theory at finite temperature and phase transitions, Helv. Phys. Acta 67 (1994) 451-583.


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) 3320-41.

Wolfdieter Lang, Rationals A(n) and more.


a(n) = numerator(A(n)) with A(n):= Catalan(n)/((n+2)*2^(2*n-1)) where Catalan(n):=A000108(n)=binomial(2*n, n)/(n+1).

a(n) = numerator(8*(2*n-1)!!/((2*(n+2))!!)) with the double factorials (2*n-1)!!:=A001147(n) (with (-1)!!:=1) and (2*n)!!:=A000165(n).


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


The denominators are given in A099399.

Sequence in context: A290235 A225825 A199927 * A272276 A271597 A272505

Adjacent sequences: A099395 A099396 A099397 * A099399 A099400 A099401




Wolfdieter Lang, Nov 10 2004



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Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)