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A050470 a(n) = Sum_{d|n, n/d == 1 (mod 4)} d^2 - Sum_{d|n, n/d == 3 (mod 4)} d^2.

%I A050470 #76 Feb 16 2025 08:32:40
%S A050470 1,4,8,16,26,32,48,64,73,104,120,128,170,192,208,256,290,292,360,416,
%T A050470 384,480,528,512,651,680,656,768,842,832,960,1024,960,1160,1248,1168,
%U A050470 1370,1440,1360,1664,1682,1536,1848,1920,1898,2112,2208,2048,2353,2604
%N A050470 a(n) = Sum_{d|n, n/d == 1 (mod 4)} d^2 - Sum_{d|n, n/d == 3 (mod 4)} d^2.
%C A050470 Number 7 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A050470 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A050470 Multiplicative because it is the Dirichlet convolution of A000290 = n^2 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - _Christian G. Bower_, May 17 2005
%H A050470 Reinhard Zumkeller, <a href="/A050470/b050470.txt">Table of n, a(n) for n = 1..10000</a>
%H A050470 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 A050470 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 A050470 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 A050470 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A050470 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%H A050470 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A050470 G.f.: Sum_{n>=1} n^2*x^n/(1+x^(2*n)). - _Vladeta Jovovic_, Oct 16 2002
%F A050470 From _Michael Somos_, Aug 08 2005: (Start)
%F A050470 Euler transform of period 4 sequence [ 4, -2, 4, -6, ...].
%F A050470 Expansion of eta(q^2)^6 * eta(q^4)^4 / eta(q)^4 in powers of q.
%F A050470 G.f.: x Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4.
%F A050470 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 8*v) * (v - 4*w) - v^2 * (v - 8*w)^2. (End)
%F A050470 G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k) / (1 - x^k)^3. - _Michael Somos_, Sep 02 2005
%F A050470 Expansion of q * phi(q)^2 * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Aug 15 2007
%F A050470 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A120030.
%F A050470 a(n) = A050461(n) - A050465(n). - _Reinhard Zumkeller_, Mar 06 2012
%F A050470 Multiplicative with a(p^e) = ((p^2)^(e+1) - Chi(p)^(e+1))/(p^2 - Chi(p)), Chi = A101455. - _Jianing Song_, Oct 30 2019
%F A050470 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/32 (A153071). - _Amiram Eldar_, Nov 04 2023
%F A050470 a(n) = Sum_{d|n} (n/d)^2*sin(d*Pi/2). - _Ridouane Oudra_, Sep 26 2024
%e A050470 G.f. = q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + 64*q^8 + ...
%t A050470 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^3 (QPochhammer[ q^4] / QPochhammer[ q])^2)^2, {q, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t A050470 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^2 / 4)^2, {q, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t A050470 a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[n/d, 2] (-1)^Quotient[n/d, 2], {d, Divisors@n}]]; (* _Michael Somos_, May 17 2015 *)
%t A050470 s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t A050470 f[p_, e_] := (p^(2*e+2) - s[p]^(e+1))/(p^2 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o A050470 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * (n/d%2) * (-1)^(n/d\2)))};
%o A050470 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * (eta(x^4 + A) / eta(x + A))^4, n))}; /* _Michael Somos_, May 17 2015 */
%o A050470 (Haskell)
%o A050470 a050470 n = a050461 n - a050465 n  -- _Reinhard Zumkeller_, Mar 06 2012
%o A050470 (Magma) Basis( ModularForms( Gamma1(4), 3), 51) [2]; /* _Michael Somos_, May 17 2015 */
%o A050470 (Python)
%o A050470 from math import prod
%o A050470 from sympy import factorint
%o A050470 def A050470(n): return prod((p**(e+1<<1)-(m:=(0,1,0,-1)[p&3]))//(p**2-m) for p, e in factorint(n).items()) # _Chai Wah Wu_, Jun 21 2024
%Y A050470 Cf. A050461, A050465, A101455, A120030, A153071.
%Y A050470 Cf. A027750, A000122, A000700, A010054, A121373.
%Y A050470 Glaisher's E'_i (i=0..12): A002654, A050469, this sequence, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.
%K A050470 nonn,easy,mult
%O A050470 1,2
%A A050470 _N. J. A. Sloane_, Dec 23 1999