OFFSET
0,5
COMMENTS
Arises from an identity of Slater's.
Number of partitions of 2*n+1 into distinct odd parts. - Vladeta Jovovic, May 08 2003
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of partitions of 2n+1 such that if k is the largest part, then k occurs an odd number of times and each integer from 1 to k-1 occurs a positive even number of times. Example: a(4)=2 because we have [3,2,2,1,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 16 2006
Difference between number of partitions of 2n+1 with an odd number of parts and those with an even number of parts (this is a consequence of Jovovic's comment above). - George Beck, May 22 2016
Let b(k) be the convolution inverse of A035457, k=1, 2, 3, ...; then a(n) = -b(4n+3), n = 0, 1, 2, 3, ... (conjectured). - George Beck, Aug 19 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities Quaestiones Math., 24 (2001), 403-416.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33--37. MR0541341 (80j:05010). See Theorem 3. [From N. J. A. Sloane, Mar 19 2012]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From N. J. A. Sloane, Dec 25 2012
FORMULA
Euler transform of period 16 sequence [ 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: ( H(sqrt(x)) - H(-sqrt(x)) ) / (2*sqrt(x)), where H(x)=prod(i>=1, 1+x^(2*i-1) ). - Emeric Deutsch, Apr 16 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 14 2015
Expansion of f(x, x^7) / f(-x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 04 2016
EXAMPLE
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + ...
G.f. = q^23 + q^71 + q^119 + q^167 + 2*q^215 + 2*q^263 + 3*q^311 + 4*q^359 + ...
MAPLE
h:=product(1+x^(2*i-1), i=1..60): hser:=series(h, x=0, 120): seq(coeff(hser, x^(2*n+1)), n=0..56); # Emeric Deutsch, Apr 16 2006
MATHEMATICA
H[x_] := x*QPochhammer[-1/x, x^2]/(1 + x); s = (H[Sqrt[x]] - H[-Sqrt[x]]) / (2*Sqrt[x]) + O[x]^60; CoefficientList[s, x] (* Jean-François Alcover, Nov 14 2015, after Emeric Deutsch *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, n=2*n+1; A = x * O(x^n); -polcoeff( eta(x + A) / eta(x^2 + A), n))}; /* Michael Somos, Jul 18 2006 */
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
N. J. A. Sloane, May 05 2002
STATUS
approved