A109506 - OEIS

%I #28 Sep 12 2023 02:24:25

%S 1,-3,4,-3,6,-12,8,-3,13,-18,12,-12,14,-24,24,-3,18,-39,20,-18,32,-36,

%T 24,-12,31,-42,40,-24,30,-72,32,-3,48,-54,48,-39,38,-60,56,-18,42,-96,

%U 44,-36,78,-72,48,-12,57,-93,72,-42,54,-120,72,-24,80,-90,60,-72,62,-96,104,-3,84,-144,68,-54,96,-144,72

%N Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.

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

%C Denoted by xi(n) in Glaisher 1907. - _Michael Somos_, May 17 2013

%D G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(18). MR0121327 (22 #12066).

%D J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 8).

%H Seiichi Manyama, <a href="/A109506/b109506.txt">Table of n, a(n) for n = 1..10000</a>

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

%H Michael Somos, <a href="/A010815/a010815.txt">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>.

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.

%F Expansion of (1 - eta(q)^8 / eta(q^2)^4) / 8 in powers of q.

%F a(n) = Sum_{d divides n} (-1)^(n/d + d) * d [Glaisher].

%F Multiplicative with a(2^e) = -3, if e>0. a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.

%F G.f.: Sum_{k>0} k * (x^k / (1 - x^k) - 6 * x^(2*k) / (1 - x^(2*k)) + 8 * x^(4*k) / (1 - x^(4*k))).

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

%F a(n) = -(-1)^n * A046897(n). a(n) = -A096727(n) / 8 unless n=0. a(2*n) = -3 * A000593(n). a(2*n + 1) = A008438(n). a(4*n + 1) = A112610(n). a(4*n + 3) = A097723(n).

%F Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * zeta(s-1) * zeta(s). - _Amiram Eldar_, Sep 12 2023

%e q - 3*q^2 + 4*q^3 - 3*q^4 + 6*q^5 - 12*q^6 + 8*q^7 - 3*q^8 + 13*q^9 + ...

%t a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ If[ Mod[ d, 4] == 0, 0, d], {d, Divisors@n}]] (* _Michael Somos_, May 17 2013 *)

%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, if( d%4, d)))}

%o (PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); -1/8 * polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))}

%Y Cf. A000593, A008438, A046897, A096727, A112610.

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

%K sign,easy,mult

%O 1,2

%A _Michael Somos_, Jun 30 2005