A122135 Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.

1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 26, 31, 40, 48, 60, 72, 89, 106, 130, 154, 186, 220, 264, 310, 370, 433, 512, 598, 704, 818, 958, 1110, 1293, 1494, 1734, 1996, 2308, 2650, 3052, 3496, 4014, 4584, 5248, 5980, 6825, 7760, 8834, 10020, 11380, 12882, 14594

Offset

0,3

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 26 2022: (Start)

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:

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

(11) (21) (22) (32) (33) (43) (44) (54)

(31) (41) (42) (52) (53) (63)

(221) (51) (61) (62) (72)

(321) (331) (71) (81)

(2211) (421) (332) (432)

(3211) (431) (441)

(521) (531)

(3311) (621)

(4211) (3321)

(4311)

(5211)

The even-length case appears to be A122134.

The odd-length case is A351595.

The alternately unequal version appears to be A122129, even A351008, odd A122130.

The alternately equal version is A351003, even A351012, odd A000009.

The alternately equal and unequal version is A351005, even A035457, odd A351593.

The alternately unequal and equal version is A351006, even A351007, odd A053251.

(End)

References

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)

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, 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]

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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.

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, ...].

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

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.

a(n) ~ (3 + sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016

Example

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 + ...
G.f. = q^9 + q^49 + 2*q^89 + 2*q^129 + 3*q^169 + 4*q^209 + 6*q^249 + ...

Maple

f:=n->1/mul(1-q^(20*k+n), k=0..20);
f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19);
series(%, q, 200); seriestolist(%); # N. J. A. Sloane, Mar 19 2012

Mathematica

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 *)
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 *)

Prog

(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))};

Crossrefs

Cf. A000009, A003242, A035363, A053251, A122129, A122130, A122134.

Sequence in context: A035939 A365827 A116665 * A339572 A027194 A039883

Adjacent sequences: A122132 A122133 A122134 * A122136 A122137 A122138

Keyword

nonn

Author

Michael Somos, Aug 21 2006

Status

approved