A138505 Expansion of ((phi(q) * phi(-q^2)^2)^2 - 1) / 4 in powers of q where phi() is a Ramanujan theta function.
1, -1, -8, -1, 26, 8, -48, -1, 73, -26, -120, 8, 170, 48, -208, -1, 290, -73, -360, -26, 384, 120, -528, 8, 651, -170, -656, 48, 842, 208, -960, -1, 960, -290, -1248, -73, 1370, 360, -1360, -26, 1682, -384, -1848, 120, 1898, 528, -2208, 8, 2353, -651, -2320
Offset: 1
Examples
G.f. = q - q^2 - 8*q^3 - q^4 + 26*q^5 + 8*q^6 - 48*q^7 - q^8 + 73*q^9 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := If[ n < 1, 0, - DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] (-1)^(n/#) &]]; (* Michael Somos, Sep 25 2015 *)
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PARI
{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d) * -(-1)^(n/d)))}; -
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))};
Formula
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).
G.f.: Sum_{k>0} -(-1)^k * (2*k-1)^2 * x^(2*k-1) / (1 + x^(2*k-1)).
a(2*n) = (-1)^n * a(n).
4 * a(n) = A138504(n) unless n=0.
a(n) = -(-1)^n * A002173(n). - Michael Somos, Sep 25 2015
Comments