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A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function. 3
1, 4, 9, 14, 16, 18, 21, 28, 36, 38, 40, 36, 43, 52, 54, 62, 56, 72, 74, 72, 81, 64, 88, 90, 98, 100, 72, 110, 112, 126, 133, 104, 126, 108, 136, 144, 112, 148, 144, 158, 144, 144, 183, 172, 180, 182, 152, 162, 194, 196, 198, 160, 216, 216, 180, 224, 189, 230 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 f(-x^2)^3 * phi(-x^3)^3 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^6 / (eta(q)^4 * eta(q^6)^3) in powers of q.

Euler transform of period 6 sequence [4, -1, -2, -1, 4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260301. - Michael Somos, Nov 13 2015

a(n) = A260109(2*n) = A263021(3*n) = A124815(4*n + 1) = A209613(4*n + 1). - Michael Somos, Nov 13 2015

a(3*n + 1) = 4 * A260165(n). a(3*n + 2) = 9 * A263021(n). - Michael Somos, Nov 13 2015

EXAMPLE

G.f. = 1 + 4*x + 9*x^2 + 14*x^3 + 16*x^4 + 18*x^5 + 21*x^6 + 28*x^7 + ...

G.f. = q + 4*q^5 + 9*q^9 + 14*q^13 + 16*q^17 + 18*q^21 + 21*q^25 + 28*q^29 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6]))^3 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^6 EllipticTheta[ 4, 0, x^3]^3 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 4, 0, x^3]^3 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^6 / (eta(x + A)^4 * eta(x^6 + A)^3), n))};

CROSSREFS

Cf. A124815, A209613, A260109 A260165, A260301, A263021.

Sequence in context: A312987 A050986 A284956 * A034259 A312988 A010452

Adjacent sequences:  A261442 A261443 A261444 * A261446 A261447 A261448

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 18 2015

STATUS

approved

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Last modified August 26 01:43 EDT 2019. Contains 326324 sequences. (Running on oeis4.)