A050468 a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.

1, 16, 80, 256, 626, 1280, 2400, 4096, 6481, 10016, 14640, 20480, 28562, 38400, 50080, 65536, 83522, 103696, 130320, 160256, 192000, 234240, 279840, 327680, 391251, 456992, 524960, 614400, 707282, 801280, 923520, 1048576, 1171200

Offset

1,2

Comments

Multiplicative because it is the Dirichlet convolution of A000583 = n^4 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

Called E'_4(n) by Hardy.

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

References

Emil Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 120.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135, section 9.3. MR0106147 (21 #4881)

Links

Seiichi Manyama, 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).

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

a(2*n + 1) = A204342(n). a(2*n) = 16 * a(n).

G.f.: Sum_{n>=1} n^4*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002

From Michael Somos, Jan 14 2012: (Start)

Expansion of eta(q^2)^2 * eta(q^4)^4 * (eta(q)^4 + 20 * eta(q^4)^8 / eta(q)^4) in powers of q.

a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4), a(p^e) = ((p^4)^(e+1) - (-1)^(e+1)) / (p^4 + 1) if p == 3 (mod 4). (End)

From Michael Somos, Jan 15 2012: (Start)

Expansion of theta_3(q^2) * (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.

Expansion of x * phi(x)^2 * (psi(x)^8 + 4 * x * psi(x^2)^8) in powers of x where phi(), psi() are Ramanujan theta functions. (End)

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^5 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204372. - Michael Somos, May 03 2015

From Amiram Eldar, Nov 04 2023: (Start)

Multiplicative with a(p^e) = (p^(4*e+4) - A101455(p)^(e+1))/(p^4 - A101455(p)).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/1536 (A175571). (End)

Example

G.f. = x + 16*x^2 + 80*x^3 + 256*x^4 + 626*x^5 + 1280*x^6 + 2400*x^7 + 4096*x^8 + ...

Mathematica

edashed[r_, n_] := Plus@@(Select[Divisors[n], Mod[n/#, 4] == 1 &]^r) - Plus@@(Select[Divisors[n], Mod[n/#, 4] == 3 &]^r); edashed[4, #] &/@Range[33] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] (EllipticTheta[ 2, 0, x]^8 + 4 EllipticTheta[ 2, 0, x^2]^8) / 256, {x, 0, 2 n}]; (* Michael Somos, Jan 11 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(4*e+4) - s[p]^(e+1))/(p^4 - 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, (n/d%2) * (-1)^((n/d - 1)/2) * d^4))}; /* Michael Somos, Sep 12 2005 */
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, n\d)))}; /* Michael Somos, Jan 14 2012 */
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^4 * (eta(x + A)^4 + 20 * x * eta(x^4 + A)^8 / eta(x + A)^4), n))}; /* Michael Somos, Jan 14 2012 */
(Magma) A := Basis( ModularForms( Gamma1(4), 5), 34); A[2] + 16*A[3]; /* Michael Somos, May 03 2015 */

Crossrefs

Cf. A000122, A000583, A000700, A010054, A101455, A121373, A175571, A204342, A204372.

Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, this sequence, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Sequence in context: A130810 A212090 A212240 * A242167 A068778 A247817

Adjacent sequences: A050465 A050466 A050467 * A050469 A050470 A050471

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Dec 23 1999

Status

approved