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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.
3
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, 32, 28, 16, 32, 32, 20, 36, 32, 28, 36, 36, 24, 32, 42, 32, 42, 40, 28, 48, 48, 32, 57, 42, 36, 48, 52, 40, 40, 64, 36, 56, 58, 32, 60, 64, 56, 64, 48, 40, 66
OFFSET
1,2
COMMENTS
Number 18 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of q * phi(q) * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^3 * eta(q^4) * eta(q^8)^2 / eta(q)^2 in powers of q.
Euler transform of period 8 sequence [ 2, -1, 2, -2, 2, -1, 2, -4, ...].
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).
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(8*k))^2.
From Peter Bala, Jan 06 2021: (Start)
a(n) = Sum_{ d | n } X(n/d)*d, where X(k) = A091337(k) is a non-principal Dirichlet charcter modulo 8.
G.f.: A(x) = Sum_{n = -oo..oo} (-1)^n*x^(4*n+1)/(1 - x^(4*n+1))^2. (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328895. - Amiram Eldar, Feb 20 2024
EXAMPLE
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 + ...
MAPLE
with(numtheory):
A091337 := n -> [0, 1, 0, -1, 0, -1, 0, 1][`mod`(n, 8)+1]:
seq(add(A091337(n/d)d, d in divisors(n)), n = 1..60); # Peter Bala, Jan 06 2021
MATHEMATICA
a[n_] := Sum[JacobiSymbol[2, d]*n/d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-François Alcover, Jan 10 2014 *)
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 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n / d * kronecker(2, d)))};
(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)))};
(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))};
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Oct 26 2006
STATUS
approved