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 A129303 Expansion of eta(q^2)^3 * eta(q^5)^2 * eta(q^10) / eta(q)^2 in powers of q. 4

%I

%S 1,2,2,4,5,4,6,8,7,10,12,8,12,12,10,16,16,14,20,20,12,24,22,16,25,24,

%T 20,24,30,20,32,32,24,32,30,28,36,40,24,40,42,24,42,48,35,44,46,32,43,

%U 50,32,48,52,40,60,48,40,60,60,40,62,64,42,64,60,48,66,64

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

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

%D S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 318 Th. 4.1

%H Vaclav Kotesovec, <a href="/A129303/b129303.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Somos, <a href="http://somos.crg4.com/multiq.html">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 Expansion of q * psi(q)^3 * psi(q^5) - q^2 * psi(q) * psi(q^5)^3 in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Jul 12 2012

%F Euler transform of period 10 sequence [ 2, -1, 2, -1, 0, -1, 2, -1, 2, -4, ...].

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

%F G.f.: Sum_{k>0} Kronecker(20, k) * x^k / (1 - x^k)^2.

%F G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3.

%F a(2*n) = a(n). a(2*n + 1) = A134080(n).

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

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 20, #] &]]; (* _Michael Somos_, Jul 12 2012 *)

%t a[ n_] := SeriesCoefficient[ (1/16) (EllipticTheta[ 2, 0, q]^3 EllipticTheta[ 2, 0, q^5] - EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^5]^3), {q, 0, 2 n}]; (* _Michael Somos_, Jul 12 2012 *)

%t nmax = 100; Rest[CoefficientList[Series[x * Product[(1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 20, d)))};

%o (PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 20, p); (p^(e+1) - f^(e+1)) / (p - f) ))};

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

%Y Cf. A134080.

%K nonn,mult

%O 1,2

%A _Michael Somos_, Apr 08 2007

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Last modified September 15 18:22 EDT 2019. Contains 327082 sequences. (Running on oeis4.)