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A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions. 1
1, 4, 4, 0, 6, 16, 8, 0, 13, 24, 12, 0, 14, 32, 24, 0, 18, 52, 20, 0, 32, 48, 24, 0, 31, 56, 40, 0, 30, 96, 32, 0, 48, 72, 48, 0, 38, 80, 56, 0, 42, 128, 44, 0, 78, 96, 48, 0, 57, 124, 72, 0, 54, 160, 72, 0, 80, 120, 60, 0, 62, 128, 104, 0, 84, 192, 68, 0, 96 (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).

LINKS

Table of n, a(n) for n=1..69.

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^8)^2 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.

Euler transform of period 8 sequence [ 4, -6, 4, 0, 4, -6, 4, -4, ...].

a(n) is multiplicative with a(2) = 4, a(2^e) = 0 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133690.

a(4*n) = 0. a(4*n+2) = 4 * sigma(2*n+1). a(2*n+1) = sigma(2*n+1).

a(n) = -(-1)^n * A121455(n). Convolution square of A113411.

a(2*n + 1) = A008438. a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).

EXAMPLE

G.f. = q + 4*q^2 + 4*q^3 + 6*q^5 + 16*q^6 + 8*q^7 + 13*q^9 + 24*q^10 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]/2)^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, if( n%2, sigma(n), if( n%4, 4 * sigma(n/2), 0)))};

(PARI) {a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3) )^2, n))};

CROSSREFS

Cf. A008438, A097723, A112160, A121455, A133690.

Sequence in context: A016705 A245592 A169783 * A121455 A227033 A285050

Adjacent sequences:  A133654 A133655 A133656 * A133658 A133659 A133660

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Sep 20 2007

STATUS

approved

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Last modified May 29 04:33 EDT 2020. Contains 334697 sequences. (Running on oeis4.)