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.

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, 211, 249, 294, 345, 404, 473, 551, 642, 747, 865, 1002, 1159, 1336, 1539, 1771, 2033, 2331, 2670, 3052, 3485, 3976, 4527, 5150, 5854, 6642, 7530, 8529, 9647, 10902

Offset: 0

Author

Michael Somos, Aug 21 2006, corrected Aug 21 2006

Keywords

• nonn
• easy

Comments

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

From Gus Wiseman, Feb 19 2022: (Start)

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:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(211) (311) (321) (322) (422) (432)

(411) (421) (431) (522)

(511) (521) (531)

(611) (621)

(711)

(32211)

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)

Wiseman's first conjecture above was proved by Connor, Proposition 2. - Peter Bala, Dec 22 2024

Examples

			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 + ...
G.f. = q^31 + q^71 + q^111 + 2*q^151 + 2*q^191 + 3*q^231 + 4*q^271 + 5*q^311 + ...
		

References

• G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(b), p. 591.
• 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)

Links

• Michael Somos, Introduction to Ramanujan theta functions
• Willard G. Connor, Partition Theorems Related to Some Identities of Rogers and Watson, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A035363, A035457, A053251, A122129, A122134, A122135, A351005, A351008.

Programs

Mathematica

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 *)
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 *)
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 *)

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))};

Formula

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

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.

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, ...].

G.f.: Sum_{k>0} x^(k^2 - 1) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2k-1))).

G.f.: 1/(Product_{k>0} (1-x^(2k-1))(1-x^(20k-8))(1-x^(20k-12))).

a(n) ~ (3-sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015