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

Author

Michael Somos, Apr 08 2007

Keywords

• nonn
• easy
• mult

Comments

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

Examples

			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 + ...
		

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.

Crossrefs

Cf. A134080.

Cf. A000122, A000700, A010054, A121373.

Programs

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 *)

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))};

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