A271661 Expansion of phi(-x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

1, 1, 2, 3, 5, 7, 9, 13, 18, 24, 32, 42, 55, 71, 91, 116, 147, 185, 231, 288, 357, 440, 540, 661, 807, 980, 1186, 1432, 1724, 2069, 2476, 2956, 3521, 4183, 4958, 5865, 6923, 8155, 9587, 11251, 13180, 15411, 17990, 20967, 24399, 28348, 32886, 38098, 44075

Offset

0,3

Comments

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

Partitions which are "4-close" to being self-conjugate; see A108961 for the definition. - Arvind Ayyer, Apr 13 2021

References

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988. See page 6 equation 2.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.

Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Wikipedia, Bailey pair

Formula

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

Expansion of q^(1/24) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 0, ...].

G.f.: Sum_{k>=0} x^(k^2) (-x, x^2)_k / (x)_{2*k}.

a(n) ~ Pi * BesselI(1, Pi*sqrt(24*n-1)/(4*sqrt(3))) / sqrt(24*n-1) ~ exp(sqrt(n/2)*Pi) / (2^(7/4)*sqrt(3)*n^(3/4)) * (1 - (3/(4*Pi) + Pi/48)/sqrt(2*n) + (5/128 - 15/(64*Pi^2) + Pi^2/9216)/n). - Vaclav Kotesovec, Apr 18 2016, extended Jan 10 2017

Example

G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 18*x^8 + ...
G.f. = q^-1 + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 9*q^143 + 13*q^167 + ..

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^6] / QPochhammer[ x], {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x, 2 k] // FunctionExpand, {k, 0, Sqrt@n}], {x, 0, n}]];
nmax = 50; CoefficientList[Series[Product[(1-x^(6*k)) / ((1-x^k) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 18 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A000700, A070047, A108961, A108962, A280937, A280938.

Sequence in context: A341497 A332686 A069999 * A035563 A240063 A216374

Adjacent sequences: A271658 A271659 A271660 * A271662 A271663 A271664

Keyword

nonn

Author

Michael Somos, Apr 11 2016

Status

approved