OFFSET
0,5
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Case k=3,i=2 of Gordon/Goellnitz/Andrews Theorem.
Also number of partitions in which no odd part is repeated, with at most 1 part of size less than or equal to 2 and where differences between parts at distance 2 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S27.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/4) * (eta(q^2) * eta(q^3) * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Jun 28 2004
Euler transform of period 12 sequence [1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, ...]. - Michael Somos, Jun 28 2004
Expansion of psi(-x^3) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 21 2007
Given g.f. A(x), then B(x) = x * A(x^4) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = u^3 * (1 + v^4) - v * (1 + u*v)^3. - Michael Somos, Nov 21 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = sqrt(1 / 3) / f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 21 2007
A101195(n) = (-1)^n * a(n).
From Vaclav Kotesovec, Jan 12 2017: (Start)
a(n) ~ Pi * BesselI(1, sqrt(4*n+1)*Pi/(2*sqrt(3))) / (3*sqrt(4*n+1)).
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)) * (1 + (Pi/24 - 3/(8*Pi))*sqrt(3/n) + (Pi^2/384 - 45/(128*Pi^2) - 15/64)/n).
(End)
EXAMPLE
1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
q + q^5 + q^9 + q^13 + 2*q^17 + 3*q^21 + 3*q^25 + 4*q^29 + 6*q^33 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k - 5))*(1 + x^(6*k - 1))/((1 - x^(6*k - 4))*(1 - x^(6*k - 2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2017 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ([1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0][(k-1)%12 + 1]) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jun 28 2004 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 28 2004 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved