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A124340 Number of solutions to n = x^2 + 2*y^2 + 4*(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.

%I A124340 #43 Aug 06 2025 10:08:30
%S A124340 1,2,2,4,4,4,8,8,7,8,10,8,12,16,8,16,18,14,18,16,16,20,24,16,21,24,20,
%T A124340 32,28,16,32,32,20,36,32,28,36,36,24,32,42,32,42,40,28,48,48,32,57,42,
%U A124340 36,48,52,40,40,64,36,56,58,32,60,64,56,64,48,40,66
%N A124340 Number of solutions to n = x^2 + 2*y^2 + 4*(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.
%C A124340 Number 18 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A124340 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A124340 G. C. Greubel, <a href="/A124340/b124340.txt">Table of n, a(n) for n = 1..10000</a>
%H A124340 David Broadhurst and Daniele Dorigoni, <a href="https://arxiv.org/abs/2507.21352">Resurgent Lambert series with characters</a>, arXiv:2507.21352 [math.NT], 2025. See p. 37.
%H A124340 Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A124340 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%H A124340 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A124340 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A124340 Expansion of q * phi(q) * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
%F A124340 Expansion of eta(q^2)^3 * eta(q^4) * eta(q^8)^2 / eta(q)^2 in powers of q.
%F A124340 Euler transform of period 8 sequence [ 2, -1, 2, -2, 2, -1, 2, -4, ...].
%F A124340 a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1, 7 (mod 8), a(p^e) = (p^(e+1) + (-1)^e)/(p + 1) if p == 3, 5 (mod 8).
%F A124340 G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
%F A124340 G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(8*k))^2.
%F A124340 From _Peter Bala_, Jan 06 2021: (Start)
%F A124340 a(n) = Sum_{ d | n } X(n/d)*d, where X(k) = A091337(k) is a non-principal Dirichlet charcter modulo 8.
%F A124340 G.f.: A(x) = Sum_{n = -oo..oo} (-1)^n*x^(4*n+1)/(1 - x^(4*n+1))^2. (End)
%F A124340 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328895. - _Amiram Eldar_, Feb 20 2024
%e A124340 G.f. = q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 4*q^6 + 8*q^7 + 8*q^8 + 7*q^9 + ...
%p A124340 with(numtheory):
%p A124340 A091337 := n -> [0, 1, 0, -1, 0, -1, 0, 1][`mod`(n, 8)+1]:
%p A124340 seq(add(A091337(n/d)d, d in divisors(n)), n = 1..60); # _Peter Bala_, Jan 06 2021
%t A124340 a[n_] := Sum[JacobiSymbol[2, d]*n/d, {d, Divisors[n]}]; a /@ Range[80] (* _Jean-François Alcover_, Jan 10 2014 *)
%t A124340 a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^3 QPochhammer[ q^4] QPochhammer[ q^8]^2 / QPochhammer[ q]^2, {q, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)
%o A124340 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n / d * kronecker(2, d)))};
%o A124340 (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( 2, p); (p^(e+1) - f^(e+1)) / (p - f)))};
%o A124340 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x + A)^2, n))};
%Y A124340 Cf. A091337, A328895.
%Y A124340 Cf. A000122, A000700, A010054, A121373.
%K A124340 nonn,mult
%O A124340 1,2
%A A124340 _Michael Somos_, Oct 26 2006