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A113428
Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.
7
1, 0, -1, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
See the Hardy-Wright reference for the identity given as g.f. formula below. - Wolfdieter Lang, Oct 28 2016
REFERENCES
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 356, p. 284.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities
FORMULA
Expansion of G(x) * f(-x) in powers of x where G() is the g.f. of A003114.
Euler transform of period 5 sequence [ 0, -1, -1, 0, -1, ...].
|a(n)| is the characteristic function of the numbers in A057569.
The exponents in the q-series q * A(q^40) are the square of the numbers in A090771.
G.f.: Sum_{k in Z} (-1)^k * x^((5*k^2 + k)/2) = Prod_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)).
Convolution of A003114 and A010815.
From Wolfdieter Lang, Oct 30 2016: (Start)
a(n) = (-1)^k if n = b(2*k+1) for k >= 0, a(n) = (-1)^k if n = b(2*k), for k >= 1, and a(n) = 0 otherwise, where b(n) = A057569(n). See the third formula.
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+1)/2)*(1-x^(2*(2*n+1))). See the Hardy reference, p. 93, eq. (6.11.1) with k=2, a=x and C_n = 1.
(End)
G.f.: Sum_{n >= 0} x^(n^2)*Product_{k >= n+1} 1 - x^k. Cf. A113429. - Peter Bala, Feb 12 2021
EXAMPLE
G.f. = 1 - x^2 - x^3 + x^9 + x^11 - x^21 - x^24 + x^38 + x^42 - x^60 - x^65 + ...
G.f. = q - q^81 - q^121 + q^361 + q^441 - q^841 - q^961 + q^1521 + q^1681 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k * [1, 0, 1, 1, 0][k%5 + 1], 1 + x * O(x^n)), n))};
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Michael Somos, Oct 31 2005
STATUS
approved