OFFSET
0,9
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
(Number of partitions of n into an even number of parts) - (number of partitions of n into an odd number of parts). [Fine]
Number 3 of the 130 identities listed in Slater 1952. - Michael Somos, Aug 20 2015
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 425, Corollary 1, Eqs. (37)-(40).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 38, Eq. (22.14).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Mansour, Toufik Counting corners in partitions. Ramanujan J. 39, No. 1, 201-224 (2016).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=0. - N. J. A. Sloane, Aug 31 2014
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. 13.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_{e-o}(n).
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp.147-167, (1952).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: Product_{k>0} (1 - x^(2*k-1)) = Product_{k>0} 1 / (1 + x^k) = 1 + Sum_{k>0} (-x)^k / (Product_{i=1..k} (1 - x^i)).
This is the convolution inverse of A000009 (partitions into distinct parts) - i.e. the negation of the INVERTi transform of A000009. - Franklin T. Adams-Watters, Jan 06 2006
Expansion of chi(-x) = chi(-x^2) / chi(x) = phi(-x) / f(-x) = phi(-x^2) / f(x) = psi(-x) / f(-x^4) = f(-x) / f(-x^2) = f(-x^2) / psi(x) in powers of x where chi(), psi(), phi(), f() are Ramanujan theta functions.
Expansion of chi(x) * chi(-x) = f(x) / psi(x) = f(-x) / psi(-x) in powers of x^2 where chi(), psi(), f() are Ramanujan theta functions.
Expansion of f(-x, -x^5) / psi(x^3) = phi(-x^3) / f(x, x^2) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jun 03 2015
Given g.f. A(x), then B(q) = A(q^3)^8 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2*v + 16*u - v^2.
G.f. A(x) satisfies A(x^2) = A(x) * A(-x).
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000009.
Euler transform of period 2 sequence [ -1, 0, ...].
Expansion of q^(1/24) * f1(t) in powers of q = exp(Pi i t) where f1() is a Weber function.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*24^(1/4)*n^(3/4)). - Vaclav Kotesovec, Feb 25 2015
a(n) = Sum_{k = 0..n} (-1)^k*A072233(n,k). - Peter Luschny, Aug 03 2015
G.f.: Sum_{k>=0} (-1)^k * x^k^2 / (Product_{i=1..k} (1 - x^(2*i))). - Michael Somos, Aug 20 2015
Sum_{j>=0} a(n-A000217(j)) = 0 if n not in A152749, = (-1)^k if n is in A152749, n=k*(3*k-1). [Merca, Corollary 4.1] - R. J. Mathar, Jun 18 2016
a(n) = -(1/n)*Sum_{k=1..n} A000593(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 28 2018
G.f.: (1 - x) * Sum_{k >= 0} (-1)^k * x^(k*(k+2)) / (Product_{i = 1..k} (1 - x^(2*i))) = (1 - x)*(1 - x^3) * Sum_{k >= 0} (-1)^k * x^(k*(k+4)) / (Product_{i = 1..k} (1 - x^(2*i)))= (1 - x)*(1 - x^3)*(1 - x^5) *
Sum_{k >= 0} (-1)^k * x^(k*(k+6)) / (Product_{i = 1..k} (1 - x^(2*i))) = .... - Peter Bala, Jan 15 2021
EXAMPLE
G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 2*x^11 + ...
G.f. = 1/q - q^23 - q^71 + q^95 - q^119 + q^143 - q^167 + 2*q^191 - 2*q^215 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[1 - x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(1/24) DedekindEta[t] / DedekindEta[2 t], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {}, x^2, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {-1}, x, -1] 2, {x, 0, n}]; (* Michael Somos, May 11 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / ( 1 - m)^2 / (16 q))^(-1/24), {q, 0, n}]]; (* Michael Somos, May 11 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A), n))};
(Sage)
from sage.combinat.partition import number_of_partitions_length
print([sum((-1)^k*number_of_partitions_length(n, k) for k in (0..n)) for n in (0..69)]) # Peter Luschny, Aug 03 2015
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 18 2003
STATUS
approved