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A138558 Expansion of eta(q^2)^7 * eta(q^5)^2 * eta(q^20)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^10)^3) in powers of q. 2
1, 2, -2, -4, 1, -4, -6, 8, 7, 2, 12, 8, -12, -12, -2, -16, -16, 14, 20, -4, 12, 24, -22, -16, 1, -24, -20, 24, 30, -4, 32, 32, -24, -32, -6, -28, -36, 40, 24, 8, 42, 24, -42, -48, 7, -44, -46, 32, 43, 2, 32, 48, -52, -40, 12, -48, -40, 60, 60, 8, 62, 64, -42 (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

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

L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 343, Eq. (3.44).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q * (f(q)^3 / chi(q)) * (f(q^5) / chi(q^5)^3) in powers of q where chi(), f() are Ramanujan theta functions.

Expansion of ( f(q)^5 / f(q^5) - f(-q^2)^5 / f(-q^10) ) / 5 in powers of q where f() is a Ramanujan theta function.

Euler transform of period 20 sequence [ 2, -5, 2, -3, 0, -5, 2, -3, 2, -4, 2, -3, 2, -5, 0, -3, 2, -5, 2, -4, ...].

a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(5^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 2, 3 (mod 5).

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138557.

G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k))^3 * (1 - x^(5*k)) * (1 - x^(10*k-5)) * (1 + x^(10*k))^2.

G.f.: Sum_{k>0} (-1)^k * ( f(5*k-1) + f(5*k-2) - f(5*k-3) - f(5*k-4) ) where f(k) := k * x^k / (1 - x^(2*k)).

a(n) = -(-1)^n * A111580(n). a(2*n) = 2 * A111580(n). - Michael Somos, Sep 08 2015

EXAMPLE

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

MATHEMATICA

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, # Mod[n/#, 2] KroneckerSymbol[ 5, #] &]]; (* Michael Somos, Sep 08 2015 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q]^5 / QPochhammer[ -q^5] - QPochhammer[ q^2]^5 / QPochhammer[ q^10]) / 5, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)

a[ n_] := SeriesCoefficient[ q QPochhammer[ -q]^3 Pochhammer[ q, -q] QPochhammer[ -q^5] QPochhammer[ q^5, -q^5]^3, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d * kronecker(5, d)))};

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^5 + A)^2 * eta(x^20 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^3), n))};

CROSSREFS

Cf. A111580, A138557.

Sequence in context: A327488 A065273 A140819 * A111580 A066202 A027420

Adjacent sequences:  A138555 A138556 A138557 * A138559 A138560 A138561

KEYWORD

sign,mult

AUTHOR

Michael Somos, Mar 24 2008, Mar 25 2008

STATUS

approved

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Last modified May 18 04:07 EDT 2021. Contains 343994 sequences. (Running on oeis4.)