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

Author

Michael Somos, Apr 11 2016

Keywords

• nonn

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

Examples

			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 + ..
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.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

Crossrefs

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

Programs

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 *)

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))};

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