OFFSET
0,4
COMMENTS
The inverted graded parafermionic partition function.
Also number of partitions of n into parts congruent to {1,3,7,9} mod 10. Also number of partitions of n into odd parts parts in which no part appears more than 4 times.
Number of partitions of n into distinct parts in which no part is a multiple of 5.
REFERENCES
T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Brooke Feigon, Truncated Theta Series Related to the Jacobi Triple Product Identity, arXiv:2401.04019 [math.CO], 2024. See page 16.
Nayandeep Deka Baruah and Abhishek Sarma, Arithmetic properties of 5-regular partitions into distinct parts, arXiv:2411.02978 [math.NT], 2024. See p. 2.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.
FORMULA
Euler transform of period 10 sequence [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...]. - Vladeta Jovovic, Aug 19 2004
Expansion of q^(1/6)eta(q^2)eta(q^5)/(eta(q)eta(q^10)) in powers of q.
Given g.f. A(x), then B(x)=(A(x^6)/x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(u^3+v^3)(1+uv)-uv(1-uv)^2. - Michael Somos, Jan 18 2005
G.f.: 1/product_{k>=1} (1-x^k+x^(2*k)-x^(3*k)+x^(4*k)) = 1/Product_{k>0} P10(x^k) where P10 is the 10th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(6*sqrt(15))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017
EXAMPLE
a(8)=4, the number of partitions into distinct parts that exclude the number 5 because we can write 8=7+1=6+2=4+3+1.
T60F = 1/q + q^5 + q^11 + 2*q^17 + 2*q^23 + 2*q^29 + 3*q^35 + 4*q^41 +...
MAPLE
series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)), k+1..150), x=0, 100);
MATHEMATICA
CoefficientList[ Series[ Product[1/(1 - x^k + x^(2k) - x^(3k) + x^(4k)), {k, 70}], {x, 0, 60}], x] (* Robert G. Wilson v, Aug 19 2004 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^5]/(QP[q]*QP[q^10])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^5+A)/eta(x+A)/eta(x^10+A), n))} /* Michael Somos, Jan 18 2005 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Noureddine Chair, Aug 18 2004
EXTENSIONS
Definition corrected by Vladeta Jovovic, Aug 19 2004
More terms from Robert G. Wilson v, Aug 19 2004
STATUS
approved