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A122373 Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q where c() is a cubic AGM theta function. 2
1, 4, 9, 16, 24, 36, 50, 64, 81, 96, 120, 144, 170, 200, 216, 256, 288, 324, 362, 384, 450, 480, 528, 576, 601, 680, 729, 800, 840, 864, 962, 1024, 1080, 1152, 1200, 1296, 1370, 1448, 1530, 1536, 1680, 1800, 1850, 1920, 1944, 2112, 2208, 2304, 2451, 2404 (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).

a(n) = n^2 and n > 0 if and only if n = 2^i * 3^j with i, j >=0 (numbers in A003586). - Michael Somos, Jun 08 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

M. D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures, arXiv:0806.3590 [math.NT], 2008-2010. See p. 15 eq. (4.21).

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

a(n) is multiplicative with a(2^e) = 4^e, a(3^e) = 9^e, a(p^e) = (p^(2*e + 2) - f^(e+1)) / (p^2 - f) where f = 1 if p == 1 (mod 6), f = -1 if p == 5 (mod 6).

Euler transform of period 6 sequence [ 4, -1, 0, -1, 4, -6, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A132000.

G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * (1 + (1+(-1)^k)/8).

G.f.: Product_{k>0} (1 - x^k) * (1 + x^(3*k)) * (1 + x^k)^5 * (1 - x^(3*k))^5.

Expansion of psi(q)^2 * psi(q^3)^2 * phi(-q^3)^3 / phi(-q) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 23 2012

Expansion of c(q) * c(q^2) * b(q^2)^2 / (9 * b(q)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 23 2012

EXAMPLE

q + 4*q^2 + 9*q^3 + 16*q^4 + 24*q^5 + 36*q^6 + 50*q^7 + 64*q^8 + 81*q^9 + 96*q^10 + ...

MATHEMATICA

terms = 50; QP = QPochhammer; s = QP[q^2]^5*QP[q^3]^4*(QP[q^6]/QP[q]^4) + O[q]^terms; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Jul 04 2017, from first formula *)

PROG

(PARI) {a(n) = local(A, p, e, f); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, p^(2*e), f =- (-1)^(p%3); (p^(2*e + 2) - f^(e+1)) / (p^2 - f)))))}

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^4, n))}

CROSSREFS

Cf. A003586, A132000.

Sequence in context: A052117 A033611 A033615 * A070458 A070457 A070456

Adjacent sequences:  A122370 A122371 A122372 * A122374 A122375 A122376

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Aug 30 2006

STATUS

approved

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Last modified January 18 04:47 EST 2019. Contains 319269 sequences. (Running on oeis4.)