A163819 - OEIS

%I #12 Mar 12 2021 22:24:46

%S 1,-1,0,1,-1,0,-2,-1,1,1,2,0,-2,2,0,1,0,-1,2,-1,0,-2,-2,0,1,2,0,-2,0,

%T 0,0,-1,0,0,2,1,-2,-2,0,1,2,0,0,2,-1,2,-2,0,3,-1,0,-2,-2,0,-2,2,0,0,2,

%U 0,0,0,-2,1,2,0,0,0,0,-2,0,-1,0,2,0,2,-4,0,0,-1,1,-2,0,0,0,0,0,-2,2,1,4,-2,0,2,-2,0,0,-3,2,1,0,0,-2,2,0

%N Expansion of (phi(q) * phi(q^10) - phi(q^2) * phi(q^5)) / 2 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="/A163819/b163819.txt">Table of n, a(n) for n = 1..1000</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) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e)/2 if p == 3, 17, 21, 27, 29, 31, 33, 39 (mod 40), a(p^e) = e+1 if p == 1, 9, 11, 19 (mod 40), a(p^e) = (-1)^e * (e+1) if p == 7, 13, 23, 37 (mod 40).

%F |a(n)| = A035180(n).

%e G.f. = q - q^2 + q^4 - q^5 - 2*q^7 - q^8 + q^9 + q^10 + 2*q^11 - 2*q^13 + ...

%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; phi[x_] := f[x, x]; A163819[n_] := SeriesCoefficient[ (phi[q]*phi[q^10] - phi[q^2]*phi[q^5])/2, {q, 0, n}]; Table[A163819[n], {n, 1, 50}] (* _G. C. Greubel_, Aug 05 2017 *)

%o (PARI) {a(n) = if( n<1, 0, (qfrep([1, 0; 0, 10], n)[n] - qfrep([2, 0; 0, 5], n)[n]))};

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-10, d)) * if( n%5, kronecker(5, n), (-1)^(0 != sum(k=0, sqrtint(n \ 50), issquare( n/5 - 10*k^2 )))))};

%Y Cf. A035180.

%K sign,mult

%O 1,7

%A _Michael Somos_, Aug 04 2009