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%I #29 Mar 14 2022 17:51:50
%S 1,0,1,1,2,2,4,4,6,7,10,11,16,18,24,28,36,42,54,62,78,91,112,130,159,
%T 184,222,258,308,356,424,488,576,664,778,894,1044,1196,1389,1590,1838,
%U 2098,2419,2754,3162,3596,4114,4668,5328,6032,6864,7760,8806,9936,11252
%N Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).
%C Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
%C In Watson 1937 page 275 he writes "Psi_0(q^{1/2},q) = prod_1^oo (1+q^{2n}) G(-q^2)" so this is the expansion in powers of q^2. - _Michael Somos_, Jun 29 2015
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
%C From _Gus Wiseman_, Feb 26 2022: (Start)
%C Conjecture: Also the number of even-length integer partitions y of n such that y_i != y_{i+1} for all even i. For example, the a(2) = 1 through a(9) = 7 partitions are:
%C (11) (21) (22) (32) (33) (43) (44) (54)
%C (31) (41) (42) (52) (53) (63)
%C (51) (61) (62) (72)
%C (2211) (3211) (71) (81)
%C (3311) (3321)
%C (4211) (4311)
%C (5211)
%C This appears to be the even-length version of A122135.
%C The odd-length version is A351595.
%C Cf. A035363, A035457, A350842, A350844, A351003-A351012, A351593.
%C (End)
%D G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(c), p. 591.
%D G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.6). MR0858826 (88b:11063)
%H G. C. Greubel, <a href="/A122134/b122134.txt">Table of n, a(n) for n = 0..1000</a>
%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 Mircea Merca, <a href="https://doi.org/10.1007/s10998-016-0180-x">From a Rogers's identity to overpartitions</a>, Periodica Mathematica Hungarica, Vol. 75, issue 2, 172-179, 2017.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H G. N. Watson <a href="https://doi.org/10.1112/plms/s2-42.1.274">The Mock Theta Functions (2)</a> Proceedings of the London Mathematical Society, s2-42: 274-304, 1937.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Euler transform of period 20 sequence [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, ...].
%F Expansion of f(x^4, x^6) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - _Michael Somos_, Jun 29 2015
%F Expansion of f(-x^2, x^3) / phi(-x^2) in powers of x where phi() is a Ramanujan theta function. - _Michael Somos_, Jun 29 2015
%F Expansion of G(-x) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - _Michael Somos_, Jun 29 2015
%F G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))).
%F Expansion of f(-x, -x^9) * f(-x^8, -x^12) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is the Ramanujan general theta function.
%F a(n) = number of partitions of n into parts that are each either == 2, 3, ..., 7 (mod 20) or == 13, 14, ..., 18 (mod 20). - _Michael Somos_, Jun 29 2015 [corrected by _Vaclav Kotesovec_, Nov 12 2016]
%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^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
%e G.f. = q + q^81 + q^121 + 2*q^161 + 2*q^201 + 4*q^241 + 4*q^281 + ...
%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / QPochhammer[ x, x, 2 k], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* _Michael Somos_, Jun 29 2015 *)
%t a[ n_] := SeriesCoefficient [ 1 / (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2]), {x, 0, n}]; (* _Michael Somos_, Jun 29 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, -x^5] QPochhammer[ -x^3, -x^5] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* _Michael Somos_, Jun 29 2015 *)
%t nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+2))*(1 - x^(20*k+3))*(1 - x^(20*k+4))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+7))*(1 - x^(20*k+13))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+16))*(1 - x^(20*k+17)) *(1 - x^(20*k+18))), {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 - x^i, 1 + x * O(x^(n -k^2-k)))), n))};
%Y Cf. A000009, A000041, A053251, A122129, A122130, A122135.
%K nonn
%O 0,5
%A _Michael Somos_, Aug 21 2006, Oct 10 2007
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