A271593 Expansion of psi(-x^3) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 44, 54, 72, 88, 115, 140, 180, 218, 276, 333, 416, 500, 618, 740, 906, 1080, 1312, 1558, 1880, 2224, 2666, 3143, 3746, 4402, 5220, 6114, 7216, 8426, 9903, 11530, 13498, 15672, 18280, 21168, 24608, 28424, 32940, 37956

Offset

0,3

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

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

Formula

Expansion of f(x, x^5) / phi(-x^2) in powers of x where phi(), f(, ) are Ramanujan theta functions.

Expansion of q^(-1/3) * eta(q^3) * eta(q^12) / (eta(q) * eta(q^6)) in powers of q.

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

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

a(n) ~ Pi * BesselI(1, Pi*sqrt(3*n+1) / sqrt(6)) / (4*sqrt(3*n+1)) ~ exp(sqrt(n/2)*Pi) / (2^(9/4)*sqrt(3)*n^(3/4)) * (1 + (Pi/6 - 3/(4*Pi))/sqrt(2*n) + (Pi^2/144 - 15/(64*Pi^2) - 5/16)/n). - Vaclav Kotesovec, Apr 18 2016, extended Jan 10 2017

Example

G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^10 + 4*q^13 + 5*q^16 + 8*q^19 + 10*q^22 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / (1-x^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^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))};

Crossrefs

Sequence in context: A303939 A326446 A323530 * A131945 A240308 A326526

Adjacent sequences: A271590 A271591 A271592 * A271594 A271595 A271596

Keyword

nonn,changed

Author

Michael Somos, Apr 10 2016

Status

approved