OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Conjecturally Sum_n a(n)q^(8n+5) equals theta series of sodalite. - Fred Lunnon, Mar 05 2015
Dickson writes that Liouville proved several related theorems about sums of triangular numbers. - Michael Somos, Feb 10 2020
REFERENCES
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. II, p. 23.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Hirschhorn, and J. Sellers, Some amazing facts about 4-cores, J. Num. Thy. 60 (1996), 51-69.
K. Ono, and L. Sze, 4-core partitions and class numbers, Acta. Arith. 80 (1997), 249-272.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
eta(32*z)^4/eta(8*z) = Sum_{x, y, z} q^(x^2+2*y^2+2*z^2), x, y, z >= 1 and odd.
From Michael Somos, Mar 24 2003: (Start)
Euler transform of period 4 sequence [1, 1, 1, -3, ...].
Expansion of q^(-5/8) * eta(q^4)^4/eta(q) in powers of q.
(End)
Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
G.f.: Product_{k>0} (1-q^(4*k))^4/(1-q^k).
Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 02 2008
EXAMPLE
G.f. = 1 + x + 2*x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...
G.f. = q^5 + q^13 + 2*q^21 + 3*q^29 + q^37 + 3*q^45 + 3*q^53 + 3*q^61 + 4*q^69 + ... ,
apparently the theta series of the sodalite net, aka edge-skeleton of space honeycomb by truncated octahedra. - Fred Lunnon, Mar 05 2015
MATHEMATICA
QP = QPochhammer; s = QP[q^4]^4/QP[q] + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Jul 26 2011, updated Nov 29 2015 *)
PROG
(PARI)
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^4 / eta(x + A), n))}; /* Michael Somos, Mar 24 2003 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Feb 11 2000
STATUS
approved