A110712 - OEIS

%I #25 Jan 28 2024 01:59:51

%S 1,-3,2,-1,5,-6,6,-7,7,-15,12,-2,12,-18,10,-9,16,-21,20,-5,12,-36,22,

%T -14,25,-36,20,-6,30,-30,32,-23,24,-48,30,-7,36,-60,24,-35,42,-36,42,

%U -12,35,-66,46,-18,43,-75,32,-12,52,-60,60,-42,40,-90,60,-10,62,-96,42,-41,60,-72,66,-16,44,-90,72,-49,72,-108

%N Expansion of eta(q)^3 * eta(q^5) * eta(q^10)^2 / eta(q^2)^2 in powers of q.

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

%H G. C. Greubel, <a href="/A110712/b110712.txt">Table of n, a(n) for n = 1..1000</a>

%H Shaun Cooper, <a href="https://doi.org/10.1007/s11139-009-9198-5">On Ramanujan's function k(q)=r(q)r^2(q^2)</a>, Ramanujan J., 20 (2009), 311-328; see p. 318, Th. 4.1.

%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 (phi(-q) * phi(-q^5)^3 - phi(-q)^3 * phi(-q^5)) / 4 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Jul 12 2012

%F Euler transform of period 10 sequence [-3, -1, -3, -1, -4, -1, -3, -1, -3, -4, ...].

%F Multiplicative with a(p^e) = p^e if p=5, a(p^e) = -(p^(e+1) - 5*(-1)^e) / (p + 1) if p=2, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 or 9 (mod 10), a(p^e) = (p^(e+1) - (-1)^e) / (p + 1) if p == 3 or 7 (mod 10).

%F G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^k)^2 = x * Product_{k>0} (1 - x^k)^3 * (1 - x^(5*k)) * (1 - x^(10*k))^2 / (1 - x^(2*k))^2.

%F a(n) = (-1)^(n+1) * A138483(n). - _Amiram Eldar_, Jan 28 2024

%e G.f. = q - 3*q^2 + 2*q^3 - q^4 + 5*q^5 - 6*q^6 + 6*q^7 - 7*q^8 + 7*q^9 - 15*q^10 + ...

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# # KroneckerSymbol[ 5, n/#] &]]; (* _Michael Somos_, Jul 12 2012 *)

%t a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5]), {q, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)

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

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^5 + A) * eta(x^10 + A)^2 / eta(x^2 + A)^2, n))};

%Y Cf. A138483.

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

%K sign,mult

%O 1,2

%A _Michael Somos_, Aug 05 2005