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

1, 4, 8, 16, 26, 32, 48, 64, 73, 104, 120, 128, 170, 192, 208, 256, 290, 292, 360, 416, 384, 480, 528, 512, 651, 680, 656, 768, 842, 832, 960, 1024, 960, 1160, 1248, 1168, 1370, 1440, 1360, 1664, 1682, 1536, 1848, 1920, 1898, 2112, 2208, 2048, 2353, 2604

Offset

1,2

Comments

Number 7 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).

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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. 4 and p. 8).

Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Index entries for sequences mentioned by Glaisher.

Formula

G.f.: Sum_{n>=1} n^2*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002

From Michael Somos, Aug 08 2005: (Start)

Euler transform of period 4 sequence [ 4, -2, 4, -6, ...].

Expansion of eta(q^2)^6 * eta(q^4)^4 / eta(q)^4 in powers of q.

G.f.: x Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4.

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)

G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k) / (1 - x^k)^3. - Michael Somos, Sep 02 2005

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

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.

a(n) = A050461(n) - A050465(n). - Reinhard Zumkeller, Mar 06 2012

Multiplicative with a(p^e) = ((p^2)^(e+1) - Chi(p)^(e+1))/(p^2 - Chi(p)), Chi = A101455. - Jianing Song, Oct 30 2019

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/32 (A153071). - Amiram Eldar, Nov 04 2023

Example

G.f. = q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + 64*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^3 (QPochhammer[ q^4] / QPochhammer[ q])^2)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^2 / 4)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
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 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
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 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * (n/d%2) * (-1)^(n/d\2)))};
(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 */
(Haskell)
a050470 n = a050461 n - a050465 n  -- Reinhard Zumkeller, Mar 06 2012
(Magma) Basis( ModularForms( Gamma1(4), 3), 51) [2]; /* Michael Somos, May 17 2015 */
(Python)
from math import prod
from sympy import factorint
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

Crossrefs

Cf. A050461, A050465, A101455, A120030, A153071.

Cf. A027750, A000122, A000700, A010054, A121373.

Glaisher's E'_i (i=0..12): A002654, A050469, this sequence, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Sequence in context: A277029 A013934 A167189 * A138501 A096296 A331243

Adjacent sequences: A050467 A050468 A050469 * A050471 A050472 A050473

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Dec 23 1999

Status

approved