A129390 Expansion of phi(x) * phi(-x^5) / (chi(-x^2) * chi(-x^10)) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 2, 1, 2, 3, 0, 0, 2, 0, 0, 4, 2, 1, 4, 2, 0, 0, 2, 0, 0, 2, 2, 3, 2, 3, 0, 0, 0, 0, 0, 2, 6, 0, 2, 4, 0, 0, 2, 0, 0, 5, 2, 0, 4, 2, 0, 0, 0, 0, 0, 2, 2, 4, 2, 2, 0, 0, 2, 0, 0, 1, 4, 1, 2, 4, 0, 0, 4, 0, 0, 4, 0, 2, 6, 2, 0, 0, 0, 0, 0, 4, 2, 0, 2, 1, 0, 0

Offset: 0

Author

Michael Somos, Apr 13 2007

Keywords

• nonn
• easy

Comments

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

Examples

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^7 + 4*x^10 + 2*x^11 + x^12 + 4*x^13 + ...
G.f. = q + 2*q^3 + q^5 + 2*q^7 + 3*q^9 + 2*q^15 + 4*q^21 + 2*q^23 + q^25 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A035710, A129391.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -20, #]&]]; (* Michael Somos, Nov 12 2015 *)

PARI

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker( -20, d)))};

PARI

{a(n) = my(A, p, e); if(n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p%20 <10, e+1, 1-e%2) ))};

PARI

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

Formula

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^5)^2 * eta(q^20) / (eta(q)^2 * eta(q^4) * eta(q^10)^2) in powers of q.

Euler transform of period 20 sequence [ 2, -2, 2, -1, 0, -2, 2, -1, 2, -2, 2, -1, 2, -2, 0, -1, 2, -2, 2, -2, ...].

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0, b(5^e) = 1, b(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), b(p^e) = (1 + (-1)^e)/2 if p == 11, 13, 17, 19 (mod 20).

G.f.: Sum_{k>0} a(k) * x^(2*k - 1) = Sum_{k>0} f(x^(2*k - 1)) where f(x) := x * (1 + x^2) * (1 + x^6) / (1 + x^10).

a(n) = (-1)^n * A129391(n). a(n) = A035710(2*n + 1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Dec 28 2023