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A113185 Expansion of (5*phi(q)*phi^3(q^5) - phi^3(q)*phi(q^5))/4 in powers of q where phi(q) is a Ramanujan theta function. 4

%I #17 Jan 28 2024 02:00:12

%S 1,1,-3,-2,1,1,6,-6,-7,7,-3,12,-2,-12,18,-2,9,-16,-21,20,1,12,-36,-22,

%T 14,1,36,-20,-6,30,6,32,-23,-24,48,-6,7,-36,-60,24,-7,42,-36,-42,12,7,

%U 66,-46,-18,43,-3,32,-12,-52,60,12,42,-40,-90,60,-2,62,-96,-42,41,-12,72,-66,-16,44,18,72,-49,-72,108,-2,20

%N Expansion of (5*phi(q)*phi^3(q^5) - phi^3(q)*phi(q^5))/4 in powers of q where phi(q) is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%D Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 249, Entry 8(ii).

%H G. C. Greubel, <a href="/A113185/b113185.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

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

%F Expansion of (eta(q^2)^5 * eta(q^10)^7)/(eta(q)*eta(q^4)*eta(q^5)^3 * eta(q^20)^3) in powers of q.

%F Euler transform of period 20 sequence [1, -4, 1, -3, 4, -4, 1, -3, 1, -8, 1, -3, 1, -4, 4, -3, 1, -4, 1, -4, ...].

%F a(n) is multiplicative with a(5^e) = 1, a(2^e) = ((-2)^(e+1)-1)/(-2-1)-2 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 9 (mod 10), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 7 (mod 10).

%F G.f.: (5phi(q) phi(q^5)^3 - phi(q)^3 phi(q^5))/4 where phi(q)=1+2(q+q^4+q^9+...).

%F a(n) = (-1)^n * A132069(n).

%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - _Amiram Eldar_, Jan 28 2024

%e 1 + q - 3*q^2 - 2*q^3 + q^4 + q^5 + 6*q^6 - 6*q^7 - 7*q^8 + 7*q^9 + ...

%t a[n_]:= SeriesCoefficient[(5*EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^5]^3 - EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, q^5])/4, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 16 2017 *)

%o (PARI) a(n)=if(n<1, n==0, (-1)^n*sumdiv(n,d, kronecker(5,d)*d*(-1)^d))

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^10+A)^7/(eta(x+A)*eta(x^4+A)*eta(x^5+A)^3*eta(x^20+A)^3), n))}

%o (PARI) {a(n)=local(A,p,e,a1); if(n<1, n==0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==5, 1, if(p>2, p*=kronecker(5,p); (p^(e+1)-1)/(p-1), a1=-3; for(i=2,e, a1=-2*a1-5);a1)))))}

%Y Cf. A132069.

%Y Cf. A000122, A000700, A010054, A121373.

%K sign,mult

%O 0,3

%A _Michael Somos_, Oct 17 2005, Mar 20 2008

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)