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

0,4

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)

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

Table of n, a(n) for n=0..53.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

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

Sequence in context: A304883 A280663 A052816 * A003073 A123946 A002569

Adjacent sequences: A122127 A122128 A122129 * A122131 A122132 A122133

Keyword

nonn

Author

Michael Somos, Aug 21 2006, corrected Aug 21 2006

Status

approved