A138483 Expansion of (phi(q)^3 * phi(q^5) - phi(q) * phi(q^5)^3) / 4 in powers of q where phi() is a Ramanujan theta function.

1, 3, 2, 1, 5, 6, 6, 7, 7, 15, 12, 2, 12, 18, 10, 9, 16, 21, 20, 5, 12, 36, 22, 14, 25, 36, 20, 6, 30, 30, 32, 23, 24, 48, 30, 7, 36, 60, 24, 35, 42, 36, 42, 12, 35, 66, 46, 18, 43, 75, 32, 12, 52, 60, 60, 42, 40, 90, 60, 10, 62, 96, 42, 41, 60, 72, 66, 16, 44

Offset: 1

Author

Michael Somos, Mar 20 2008

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A110712, A113185.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 25, #] (-1)^Quotient[# + 2, 5] If[ Mod[#, 4] > 0, 1, 5] &]]; (* Michael Somos, Sep 27 2015 *)
a[ n_] := SeriesCoefficient[ q EllipticTheta[3, 0, q] QPochhammer[ -q, q^2] QPochhammer[ -q^5] QPochhammer[ q^10]^2, {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, q]^3 EllipticTheta[3, 0, q^5] - EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5]^3) / 4, {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, n/d * kronecker(25, d) * (-1)^((d+2) \ 5) * if(d%4, 1, 5)))};

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) - 5*(-1)^e) / 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^2 + A)^7 * eta(x^10 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of q * phi(q) * chi(q) * f(q^5) * f(-q^10)^2 in powers of q where phi(), chi(), f() are Ramanujan theta functions.

Expansion of eta(q^2)^7 * eta(q^10)^5 / (eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [3, -4, 3, -1, 4, -4, 3, -1, 3, -8, 3, -1, 3, -4, 4, -1, 3, -4, 3, -4, ...].

a(n) is multiplicative with a(2^e) = (2^(e+1) - 5*(-1)^e) / 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).

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 80^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113185.

G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^3 * (1 - x^(5*k))^3 * (1 + x^(10*k-5))^4 * (1 + x^(10*k))^3.

a(n) = -(-1)^n * A110712(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(5*sqrt(5)) = 0.882764... . - Amiram Eldar, Nov 24 2023