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A214361 Expansion of c(q^2) * (c(q) + 2 * c(q^4)) / 9 in powers of q where c() is a cubic AGM theta function. 2
1, 3, 3, 3, 6, 9, 8, 3, 9, 18, 12, 9, 14, 24, 18, 3, 18, 27, 20, 18, 24, 36, 24, 9, 31, 42, 27, 24, 30, 54, 32, 3, 36, 54, 48, 27, 38, 60, 42, 18, 42, 72, 44, 36, 54, 72, 48, 9, 57, 93, 54, 42, 54, 81, 72, 24, 60, 90, 60, 54, 62, 96, 72, 3, 84, 108, 68, 54 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

b(n) = 6*a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z is not divisible by 3. - Michael Somos, Jun 23 2018

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (phi(q)^4 - phi(q^3)^4) / 8 = q * phi(q^3) * (chi(q) * psi(-q^3))^3 = q * psi(-q^3)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of eta(q^2)^6 * eta(q^3) * eta(q^6)^2 * eta(q^12) / ( eta(q) * eta(q^4))^3 in powers of q.

Euler transform of period 12 sequence [3, -3, 2, 0, 3, -6, 3, 0, 2, -3, 3, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A214456.

a(n) is multiplicative with a(2^e) = 3 if e>0, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

G.f.: x * Product_{k>0} (1 + (-x)^(3*k)) * (1 - x^(6*k))^4 / ( 1 + (-x)^k)^3.

a(n) = -(-1)^n * A124449(n). a(3*n) = 3*a(n). a(2*n) = a(4*n) = 3 * A121443(n). a(2*n + 1) = A185717(n).

EXAMPLE

G.f. = q + 3*q^2 + 3*q^3 + 3*q^4 + 6*q^5 + 9*q^6 + 8*q^7 + 3*q^8 + 9*q^9 + 18*q^10 + ...

a(1) = 1, b(1) = 6 with solutions (w, x, y, z) = {(0, 0, 1, 0), {0, 1, 0, 0), (1, 0, 0, 0)} and their negatives. - Michael Somos, Jun 23 2018

MATHEMATICA

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - EllipticTheta[ 3, 0, q^3]^4) / 8, {q, 0, n}];

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(3/2)]^4 / (-16 (-q)^(1/2)), {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};

(PARI) {a(n) = if( n<1, 0, sigma(n) + if( n%3==0, -1 * sigma(n/3)) + if( n%4==0, -4 * sigma(n/4)) + if( n%12==0, +4 * sigma(n/12)))};

(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 3, p==3, 3^e, (p^(e+1) - 1) / (p - 1))))};

CROSSREFS

Cf. A121443, A124449, A185717, A214456.

Sequence in context: A025497 A031502 A153004 * A124449 A262877 A141094

Adjacent sequences:  A214358 A214359 A214360 * A214362 A214363 A214364

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Jul 18 2012

STATUS

approved

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Last modified September 27 03:05 EDT 2021. Contains 347673 sequences. (Running on oeis4.)