|
%I #56 Nov 05 2023 15:56:41
%S 1,16,80,256,626,1280,2400,4096,6481,10016,14640,20480,28562,38400,
%T 50080,65536,83522,103696,130320,160256,192000,234240,279840,327680,
%U 391251,456992,524960,614400,707282,801280,923520,1048576,1171200
%N a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.
%C 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
%C Called E'_4(n) by Hardy.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D Emil Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 120.
%D 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)
%H Seiichi Manyama, <a href="/A050468/b050468.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&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>, 2019.
%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 a(2*n + 1) = A204342(n). a(2*n) = 16 * a(n).
%F G.f.: Sum_{n>=1} n^4*x^n/(1+x^(2*n)). - _Vladeta Jovovic_, Oct 16 2002
%F From _Michael Somos_, Jan 14 2012: (Start)
%F 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.
%F 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)
%F From _Michael Somos_, Jan 15 2012: (Start)
%F Expansion of theta_3(q^2) * (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.
%F 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)
%F 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
%F From _Amiram Eldar_, Nov 04 2023: (Start)
%F Multiplicative with a(p^e) = (p^(4*e+4) - A101455(p)^(e+1))/(p^4 - A101455(p)).
%F Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/1536 (A175571). (End)
%e G.f. = x + 16*x^2 + 80*x^3 + 256*x^4 + 626*x^5 + 1280*x^6 + 2400*x^7 + 4096*x^8 + ...
%t 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 *)
%t 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 *)
%t s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t 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 *)
%o (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 */
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, n\d)))}; /* _Michael Somos_, Jan 14 2012 */
%o (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 */
%o (Magma) A := Basis( ModularForms( Gamma1(4), 5), 34); A[2] + 16*A[3]; /* _Michael Somos_, May 03 2015 */
%Y Cf. A000122, A000583, A000700, A010054, A101455, A121373, A175571, A204342, A204372.
%Y Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, this sequence, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_, Dec 23 1999
|
