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

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.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

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

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: A057886 A200672 A069999 * A035563 A240063 A216374

Adjacent sequences:  A271658 A271659 A271660 * A271662 A271663 A271664

KEYWORD

nonn

AUTHOR

Michael Somos, Apr 11 2016

STATUS

approved

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Last modified February 23 23:11 EST 2018. Contains 299595 sequences. (Running on oeis4.)