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A193826 Expansion of psi(x^2) * phi(x^7) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions. 2
1, 1, 3, 4, 7, 10, 17, 26, 38, 57, 81, 114, 161, 224, 309, 419, 569, 759, 1011, 1336, 1757, 2296, 2981, 3855, 4956, 6344, 8087, 10272, 12994, 16367, 20561, 25723, 32086, 39902, 49484, 61182, 75439, 92791, 113821, 139294, 170073, 207187, 251853 (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

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of chi(-x^14)^2 / (chi(-x) * chi(-x^2)^4 * chi(-x^7)^3 ) in powers of x where chi() is a Ramanujan theta function.

Expansion of q^(1/12) * eta(q^4)^2 * eta(q^14)^5 / (eta(q) * eta(q^2) * eta(q^7)^3* eta(q^28)^2) in powers of q.

Euler transform of period 28 sequence [ 1, 2, 1, 0, 1, 2, 4, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 4, 2, 1, 0, 1, 2, 1, 0, ...].

a(n) = A102314(4*n).

a(n) ~ exp(4*Pi*sqrt(n/21)) / (2^(5/2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

EXAMPLE

G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 26*x^7 + 38*x^8 + ...

G.f. = 1/q + q^11 + 3*q^23 + 4*q^35 + 7*q^47 + 10*q^59 + 17*q^71 + 26*q^83 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, 4 n}];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^7] EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x] QPochhammer[ x^7]), {x, 0, n}];

a[ n_] := SeriesCoefficient[ QPochhammer[ x^14, x^28]^2 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2 QPochhammer[ x^7, x^14]^3), {x, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^14 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^7 + A)^3* eta(x^28 + A)^2), n))};

CROSSREFS

Cf. A102314.

Sequence in context: A261037 A280423 A143607 * A032715 A293276 A291609

Adjacent sequences:  A193823 A193824 A193825 * A193827 A193828 A193829

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 06 2011

STATUS

approved

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Last modified February 21 01:29 EST 2019. Contains 320364 sequences. (Running on oeis4.)