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A122130 Expansion of f(-x^4, -x^16) / psi(-x) in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

%I A122130 #31 Feb 16 2025 08:33:02
%S A122130 1,1,1,2,2,3,4,5,7,9,11,14,18,22,27,34,41,50,61,73,88,106,126,150,179,
%T A122130 211,249,294,345,404,473,551,642,747,865,1002,1159,1336,1539,1771,
%U A122130 2033,2331,2670,3052,3485,3976,4527,5150,5854,6642,7530,8529,9647,10902
%N A122130 Expansion of f(-x^4, -x^16) / psi(-x) in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
%C A122130 Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
%C A122130 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A122130 From _Gus Wiseman_, Feb 19 2022: (Start)
%C A122130 This appears to be the number of odd-length alternately strict integer partitions of n + 1, i.e., partitions y such that y_i != y_{i+1} for all odd i. For example, the a(1) = 1 through a(9) = 7 partitions are:
%C A122130   (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)
%C A122130                  (211)  (311)  (321)  (322)  (422)  (432)
%C A122130                                (411)  (421)  (431)  (522)
%C A122130                                       (511)  (521)  (531)
%C A122130                                              (611)  (621)
%C A122130                                                     (711)
%C A122130                                                     (32211)
%C A122130 The even-length version is A351008. Including even-length partitions appears to give A122129. Swapping strictly and weakly decreasing relations gives A351595. The constant instead of strict version is A351594. (End)
%C A122130 Wiseman's first conjecture above was proved by Connor, Proposition 2. - _Peter Bala_, Dec 22 2024
%D A122130 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(b), p. 591.
%D A122130 G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.8). MR0858826 (88b:11063)
%H A122130 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A122130 Willard G. Connor, <a href="https://doi.org/10.2307/1997097">Partition Theorems Related to Some Identities of Rogers and Watson</a>, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
%H A122130 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A122130 Expansion of f(x, x^9) / f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Nov 12 2016
%F A122130 Expansion of f(-x^2) * f(-x^20) / (f(-x) * f(-x^8, -x^12)) in powers of x where f(-x) : = f(-x, -x^2) and f(, ) is Ramanujan's general theta function.
%F A122130 Euler transform of period 20 sequence [ 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, ...].
%F A122130 G.f.: Sum_{k>0} x^(k^2 - 1) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2k-1))).
%F A122130 G.f.: 1/(Product_{k>0} (1-x^(2k-1))(1-x^(20k-8))(1-x^(20k-12))).
%F A122130 a(n) ~ (3-sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015
%e A122130 G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 9*x^9 + ...
%e A122130 G.f. = q^31 + q^71 + q^111 + 2*q^151 + 2*q^191 + 3*q^231 + 4*q^271 + 5*q^311 + ...
%t A122130 nmax = 100; CoefficientList[Series[Product[1/((1-x^(2*k-1))*(1-x^(20*k-8))*(1-x^(20*k-12))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)
%t A122130 a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[x, x^2] QPochhammer[x^8, x^20] QPochhammer[x^12, x^20]), {x, 0, n}]; (* _Michael Somos_, Nov 12 2016 *)
%t A122130 a[ n_] := SeriesCoefficient[ Sqrt[2] x^(1/8) QPochhammer[ x^4, x^20] QPochhammer[ x^16, x^20] QPochhammer[x^20] / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}] // Simplify; (* _Michael Somos_, Nov 12 2016 *)
%o A122130 (PARI) {a(n) = if( n<1, n==0, polcoeff( sum(k=1, sqrtint(n+1), x^(k^2-1) / prod(i=1, 2*k-1, 1 - x^i, 1 + x * O(x^(n-k^2+1)))), n))};
%Y A122130 Cf. A035363, A035457, A053251, A122129, A122134, A122135, A351005, A351008.
%K A122130 nonn,easy
%O A122130 0,4
%A A122130 _Michael Somos_, Aug 21 2006, corrected Aug 21 2006