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%I #29 Mar 14 2022 17:51:26
%S 1,1,2,2,3,4,6,7,10,12,16,20,26,31,40,48,60,72,89,106,130,154,186,220,
%T 264,310,370,433,512,598,704,818,958,1110,1293,1494,1734,1996,2308,
%U 2650,3052,3496,4014,4584,5248,5980,6825,7760,8834,10020,11380,12882,14594
%N Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.
%C Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C From _Gus Wiseman_, Feb 26 2022: (Start)
%C Conjecture: Also the number of integer partitions y of n such that y_i > y_{i+1} for all even i. For example, the a(1) = 1 through a(9) = 12 partitions are:
%C (1) (2) (3) (4) (5) (6) (7) (8) (9)
%C (11) (21) (22) (32) (33) (43) (44) (54)
%C (31) (41) (42) (52) (53) (63)
%C (221) (51) (61) (62) (72)
%C (321) (331) (71) (81)
%C (2211) (421) (332) (432)
%C (3211) (431) (441)
%C (521) (531)
%C (3311) (621)
%C (4211) (3321)
%C (4311)
%C (5211)
%C The even-length case appears to be A122134.
%C The odd-length case is A351595.
%C The alternately unequal version appears to be A122129, even A351008, odd A122130.
%C The alternately equal version is A351003, even A351012, odd A000009.
%C The alternately equal and unequal version is A351005, even A035457, odd A351593.
%C The alternately unequal and equal version is A351006, even A351007, odd A053251.
%C (End)
%D G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.5). MR0858826 (88b:11063)
%D G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(d), p. 591.
%H G. C. Greubel, <a href="/A122135/b122135.txt">Table of n, a(n) for n = 0..1000</a>
%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 2. [From _N. J. A. Sloane_, Mar 19 2012]
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(x^2, x^8) / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Nov 12 2016
%F Expansion of f(-x^3, -x^7) * f(-x^4, -x^16) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is Ramanujan's general theta function.
%F Euler transform of period 20 sequence [ 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, ...].
%F G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1))).
%F Let f(n) = 1/Product_{k >= 0} (1-q^(20k+n)). Then g.f. is f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19); - _N. J. A. Sloane_, Mar 19 2012.
%F a(n) ~ (3 + sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - _Vaclav Kotesovec_, Nov 12 2016
%e G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
%e G.f. = q^9 + q^49 + 2*q^89 + 2*q^129 + 3*q^169 + 4*q^209 + 6*q^249 + ...
%p f:=n->1/mul(1-q^(20*k+n),k=0..20);
%p f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19);
%p series(%,q,200); seriestolist(%); # _N. J. A. Sloane_, Mar 19 2012
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^5] QPochhammer[ x^4, -x^5] QPochhammer[-x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* _Michael Somos_, Nov 12 2016 *)
%t nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+1))*(1 - x^(20*k+2))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+8))*(1 - x^(20*k+9))*(1 - x^(20*k+11))*(1 - x^(20*k+12))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+18))*(1 - x^(20*k+19)) ), {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 12 2016 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1) - 1) \2, x^(k^2 + k) / prod(i=1, 2*k+1, 1 - x^i, 1 + x * O(x^(n-k^2-k)))), n))};
%Y Cf. A000009, A003242, A035363, A053251, A122129, A122130, A122134.
%K nonn
%O 0,3
%A _Michael Somos_, Aug 21 2006
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