A134080 Expansion of (f(-q^5)^5 / f(-q) + f(q^5)^5 / f(q)) / 2 in powers of q^2 where f() is a Ramanujan theta function.

1, 2, 5, 6, 7, 12, 12, 10, 16, 20, 12, 22, 25, 20, 30, 32, 24, 30, 36, 24, 42, 42, 35, 46, 43, 32, 52, 60, 40, 60, 62, 42, 60, 66, 44, 72, 72, 50, 72, 80, 61, 82, 80, 60, 90, 72, 64, 100, 96, 84, 102, 102, 60, 106, 110, 72, 112, 110, 84, 96, 133, 84, 125, 126

Offset: 0

Author

Michael Somos, Oct 07 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 + 5*x^2 + 6*x^3 + 7*x^4 + 12*x^5 + 12*x^6 + 10*x^7 + 16*x^8 + ...
G.f. = q + 2*q^3 + 5*q^5 + 6*q^7 + 7*q^9 + 12*q^11 + 12*q^13 + 10*q^15 + 16*q^17 + ...
		

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. A053723, A110712, A129303, A138483, A138512, A138557, A328717.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ m/d KroneckerSymbol[ 5, d], {d, Divisors @ m}]]]; (* Michael Somos, Jun 14 2014 *)

PARI

{a(n) = if( n<0, 0, n = 2*n + 1 ; sumdiv(n, d, kronecker( 5, d) * n / d)) };

Formula

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

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).

a(n) = A053723(2*n) = A110712(2*n + 1) = A129303(2*n + 1) = A138483(2*n + 1) = A138512(2*n + 1) = A138557(2*n + 1).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (5/2) * A328717 = 2*Pi^2/(5*sqrt(5)) = 1.7655285081... . - Amiram Eldar, Nov 23 2023