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A260649 Expansion of (phi(q^3) * phi(q^5) + phi(q) * phi(q^15)) / 2 - 1 in powers of q where phi(q) is a Ramanujan theta function.

%I A260649 #22 Feb 16 2025 08:33:26
%S A260649 1,0,1,1,1,0,0,2,1,0,0,1,0,0,1,3,2,0,2,1,0,0,2,2,1,0,1,0,0,0,2,4,0,0,
%T A260649 0,1,0,0,0,2,0,0,0,0,1,0,2,3,1,0,2,0,2,0,0,0,2,0,0,1,2,0,0,5,0,0,0,2,
%U A260649 2,0,0,2,0,0,1,2,0,0,2,3,1,0,2,0,2,0,0
%N A260649 Expansion of (phi(q^3) * phi(q^5) + phi(q) * phi(q^15)) / 2 - 1 in powers of q where phi(q) is a Ramanujan theta function.
%C A260649 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260649 G. C. Greubel, <a href="/A260649/b260649.txt">Table of n, a(n) for n = 1..10000</a>
%H A260649 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260649 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260649 Expansion of (eta(q^2)^2 * eta(q^6) * eta(q^10) * eta(q^30)^2) / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60)) - 1 in powers of q.
%F A260649 a(n) is multiplicative with a(2^e) = |e-1|, a(3^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1 + (-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).
%F A260649 Moebius transform of a period 60 sequence.
%F A260649 G.f.: Sum_{k>0} Kronecker(-15, k) x^k / (1 - (-x)^k).
%F A260649 a(n) = A122855(n) unless n=0.
%F A260649 a(3*n) = a(5*n) = a(n). a(4*n) = A035175(n). a(4*n + 2) = 0.
%F A260649 a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.
%e A260649 G.f. = x + x^3 + x^4 + x^5 + 2*x^8 + x^9 + x^12 + x^15 + 3*x^16 + 2*x^17 + ...
%t A260649 a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -15, #] If[ Mod[#, 4] == 2, -1, 1] &]];
%t A260649 a[ n_] := If[ n < 1, 0, Times@@ (Which[# == 1, 1, # == 2, #2 - 1, # < 6, 1, KroneckerSymbol[#, -15] == 1, #2 + 1, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])];
%t A260649 a[ n_] := SeriesCoefficient[QPochhammer[ q^2]^2 QPochhammer[ q^6] QPochhammer[ q^10] QPochhammer[ q^30]^2 / (QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^15] QPochhammer[ q^60]) - 1, {q, 0, n}];
%o A260649 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-15, d) * (-1)^(d%4==2) ))};
%o A260649 (PARI) {a(n) = if( n<1, 0, qfrep( [1, 0; 0, 15], n)[n] + qfrep( [3, 0; 0, 5], n)[n] )};
%o A260649 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, e-1, p==3 || p==5, 1, kronecker(p, -15) == 1, e+1, 1-e%2 )))};
%o A260649 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A)) - 1, n))};
%Y A260649 Cf. A035175, A122855.
%K A260649 nonn,mult
%O A260649 1,8
%A A260649 _Michael Somos_, Nov 12 2015