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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.

%I A134080 #18 Feb 16 2025 08:33:06
%S A134080 1,2,5,6,7,12,12,10,16,20,12,22,25,20,30,32,24,30,36,24,42,42,35,46,
%T A134080 43,32,52,60,40,60,62,42,60,66,44,72,72,50,72,80,61,82,80,60,90,72,64,
%U A134080 100,96,84,102,102,60,106,110,72,112,110,84,96,133,84,125,126
%N 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.
%C A134080 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A134080 G. C. Greubel, <a href="/A134080/b134080.txt">Table of n, a(n) for n = 0..10000</a>
%H A134080 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A134080 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A134080 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.
%F A134080 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).
%F A134080 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).
%F A134080 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
%e A134080 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 + ...
%e A134080 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 + ...
%t A134080 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 *)
%o A134080 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1 ; sumdiv(n, d, kronecker( 5, d) * n / d)) };
%Y A134080 Cf. A053723, A110712, A129303, A138483, A138512, A138557, A328717.
%Y A134080 Cf. A000122, A000700, A010054, A121373.
%K A134080 nonn,easy
%O A134080 0,2
%A A134080 _Michael Somos_, Oct 07 2007