A138505 - OEIS

%I #11 Mar 12 2021 22:24:45

%S 1,-1,-8,-1,26,8,-48,-1,73,-26,-120,8,170,48,-208,-1,290,-73,-360,-26,

%T 384,120,-528,8,651,-170,-656,48,842,208,-960,-1,960,-290,-1248,-73,

%U 1370,360,-1360,-26,1682,-384,-1848,120,1898,528,-2208,8,2353,-651,-2320

%N Expansion of ((phi(q) * phi(-q^2)^2)^2 - 1) / 4 in powers of q where phi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A138505/b138505.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), a(p^e) = ((-p^2)^(e+1) - 1) / ( -p^2 - 1) if p == 3 (mod 4).

%F G.f.: Sum_{k>0} -(-1)^k * (2*k-1)^2 * x^(2*k-1) / (1 + x^(2*k-1)).

%F a(2*n) = (-1)^n * a(n).

%F 4 * a(n) = A138504(n) unless n=0.

%F a(n) = -(-1)^n * A002173(n). - _Michael Somos_, Sep 25 2015

%e G.f. = q - q^2 - 8*q^3 - q^4 + 26*q^5 + 8*q^6 - 48*q^7 - q^8 + 73*q^9 + ...

%t a[ n_] := If[ n < 1, 0, - DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] (-1)^(n/#) &]]; (* _Michael Somos_, Sep 25 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d) * -(-1)^(n/d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4))^2 - 1) / 4, n))};

%Y Cf. A002173, A138504.

%K sign,mult

%O 1,3

%A _Michael Somos_, Mar 21 2008