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A271593 Expansion of psi(-x^3) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions. 1
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 (list; graph; refs; listen; history; text; internal format)
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.

M. 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

AUTHOR

Michael Somos, Apr 10 2016

STATUS

approved

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Last modified December 16 01:32 EST 2019. Contains 330013 sequences. (Running on oeis4.)