%I #59 Aug 24 2017 13:37:43
%S 1,1,1,1,2,2,3,4,5,6,8,9,12,14,17,20,25,29,35,41,49,57,68,78,93,107,
%T 125,144,168,192,223,255,294,335,385,437,501,568,647,732,833,939,1065,
%U 1199,1355,1523,1717,1925,2166,2425,2720,3040,3405,3797,4244,4727,5272
%N Expansion of Product_{i in A069909} 1/(1 - x^i).
%C Arises from an identity of Slater's.
%C Number of partitions of 2*n+1 into distinct odd parts. - _Vladeta Jovovic_, May 08 2003
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A069911/b069911.txt">Table of n, a(n) for n = 0..1000</a>
%H G. E. Andrews et al., <a href="http://dx.doi.org/10.1080/16073606.2001.9639228">q-Engel series expansions and Slater's identities</a> Quaestiones Math., 24 (2001), 403-416.
%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/0097-3165(79)90005-0">Some partition theorems of the Rogers-Ramanujan type</a>, 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]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H M. P. Zaletel and R. S. K. Mong, <a href="http://arxiv.org/abs/1208.4862">Exact Matrix Product States for Quantum Hall Wave Functions</a>, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012
%F a(n) = A000700(2n+1) = -A081362(2n+1).
%F 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
%F 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
%F a(n) ~ exp(Pi*sqrt(n/3)) / (2^(5/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 14 2015
%F Expansion of f(x, x^7) / f(-x^2) where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Jun 04 2016
%e 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 + ...
%e G.f. = q^23 + q^71 + q^119 + q^167 + 2*q^215 + 2*q^263 + 3*q^311 + 4*q^359 + ...
%p 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
%t 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_ *)
%o (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 */
%Y Cf. A000700, A035457, A069908, A069909, A069910, A081362, A122129, A122135.
%K nonn
%O 0,5
%A _N. J. A. Sloane_, May 05 2002