login
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.
3
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
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
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
EXAMPLE
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 + ...
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 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1 ; sumdiv(n, d, kronecker( 5, d) * n / d)) };
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Oct 07 2007
STATUS
approved