A138512 Expansion of q * f(q^5)^5 / f(q) in powers of q where f() is a Ramanujan theta function.

1, -1, 2, -3, 5, -2, 6, -5, 7, -5, 12, -6, 12, -6, 10, -11, 16, -7, 20, -15, 12, -12, 22, -10, 25, -12, 20, -18, 30, -10, 32, -21, 24, -16, 30, -21, 36, -20, 24, -25, 42, -12, 42, -36, 35, -22, 46, -22, 43, -25, 32, -36, 52, -20, 60, -30, 40, -30, 60, -30, 62

Offset

1,3

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

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of eta(q) * eta(q^4) * eta(q^10)^15 / (eta(q^2)^3 * eta(q^5)^5 * eta(q^20)^5) in powers of q.

Euler transform of period 20 sequence [ -1, 2, -1, 1, 4, 2, -1, 1, -1, -8, -1, 1, -1, 2, 4, 1, -1, 2, -1, -4, ...].

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

a(n) = -(-1)^n * A053723(n-1).

From Michael Somos, Sep 25 2015: (Start)

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (16/5)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138506.

a(2*n + 1) = A134080(n). (End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = A328717 / 8 = 0.0882764... . - Amiram Eldar, Nov 23 2023

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5]^5 / QPochhammer[ -q], {q, 0, n}]; (* Michael Somos, Sep 25 2015 *)
a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ n/d KroneckerSymbol[ 5, d], { d, Divisors @ n}]]; (* Michael Somos, Sep 25 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, n/d)))};
(PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(2^(e+1) - (-1)^(e+1)) / 3, f = kronecker(5, p); (p^(e+1) - f^(e+1)) / (p - f) ))) };
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^15 / (eta(x^2 + A)^3 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))};

Crossrefs

Cf. A053723, A134080, A138506, A328717.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A297996 A239692 A126833 * A053723 A201652 A359609

Adjacent sequences: A138509 A138510 A138511 * A138513 A138514 A138515

Keyword

sign,easy,mult

Author

Michael Somos, Mar 21 2008

Status

approved