A129303 Expansion of eta(q^2)^3 * eta(q^5)^2 * eta(q^10) / eta(q)^2 in powers of q.

1, 2, 2, 4, 5, 4, 6, 8, 7, 10, 12, 8, 12, 12, 10, 16, 16, 14, 20, 20, 12, 24, 22, 16, 25, 24, 20, 24, 30, 20, 32, 32, 24, 32, 30, 28, 36, 40, 24, 40, 42, 24, 42, 48, 35, 44, 46, 32, 43, 50, 32, 48, 52, 40, 60, 48, 40, 60, 60, 40, 62, 64, 42, 64, 60, 48, 66, 64

Offset

1,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; ResearchGate link. See p. 318, Th. 4.1.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of q * psi(q)^3 * psi(q^5) - q^2 * psi(q) * psi(q^5)^3 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Jul 12 2012

Euler transform of period 10 sequence [ 2, -1, 2, -1, 0, -1, 2, -1, 2, -4, ...].

a(n) is multiplicative with a(p^e) = p^e if p = 2 or 5, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).

G.f.: Sum_{k>0} Kronecker(20, k) * x^k / (1 - x^k)^2.

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

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(5*sqrt(5)) = 0.882764... . - Amiram Eldar, Dec 22 2023

Example

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

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 20, #] &]]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (1/16) (EllipticTheta[ 2, 0, q]^3 EllipticTheta[ 2, 0, q^5] - EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^5]^3), {q, 0, 2 n}]; (* Michael Somos, Jul 12 2012 *)
nmax = 100; Rest[CoefficientList[Series[x * Product[(1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 20, 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, ]; f = kronecker( 20, 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^2 + A)^3 * eta(x^5 + A)^2 * eta(x^10 + A) / eta(x + A)^2, n))};

Crossrefs

Cf. A134080.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A293974 A346036 A138557 * A255368 A186101 A284722

Adjacent sequences: A129300 A129301 A129302 * A129304 A129305 A129306

Keyword

nonn,easy,mult

Author

Michael Somos, Apr 08 2007

Status

approved