OFFSET
1,3
COMMENTS
Multiplicative because it is the Inverse Moebius transform of [1, 0, -3^2, 0, 5^2, 0, -7^2, ...], which is multiplicative. - Christian G. Bower, May 18 2005
REFERENCES
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
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).
Frazer Jarvis and Helena A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J. (22) (2010), 171.
Jan Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.
FORMULA
A120030(n) = -4*a(n), if n>0.
Multiplicative with a(p^e) = 1 if p = 2; ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4); ((-p^2)^(e+1)-1)/(-p^2-1) if p == 3 (mod 4). - David W. Wilson, Sep 01 2001 [This can be written as a single formula: a(p^e) = ((p^2*Chi(p))^(e+1) - 1)/(p^2*Chi(p) - 1), Chi = A101455. - Jianing Song, Oct 30 2019]
G.f.: Sum_{n>=1} A056594(n-1)*n^2*q^n/(1-q^n).
Expansion of (1 - theta_4(q)^2 * theta_4(q^2)^4)/4 in powers of q. - Michael Somos, Aug 09 2006
Expansion of (1-eta(q)^4*eta(q^2)^6/eta(q^4)^4)/4 in powers of q.
G.f.: q*G'(q)/G(q), with G(q) = Product_{n>=1} (1-q^n)^(4n*A056594(n+1)).
a(n) = Sum_{d|n} d^2*sin(d*Pi/2). - Ridouane Oudra, Feb 21 2023
G.f.: Sum_{n>=0} (4*n + 1)^2*x^(4*n + 1)/(1 - x^(4*n + 1)) - (4*n + 3)^2*x^(4*n + 3)/(1 - x^(4*n + 3)). - Miles Wilson, Oct 26 2024
EXAMPLE
The divisors of 15 are 1,3,5,15, so a(15)=(1^2+5^2)-(3^2+15^2) = -208.
G.f. = x + x^2 - 8*x^3 + x^4 + 26*x^5 - 8*x^6 - 48*x^7 + x^8 + 73*x^9 + ... - Michael Somos, Jun 25 2019
MAPLE
with(numtheory):
A002173:= proc(n)
local count1, count3, d;
count1 := 0:
count3 := 0:
for d in numtheory[divisors](n) do
if d mod 4 = 1 then
count1 := count1+d^2
elif d mod 4 = 3 then
count3 := count3+d^2
fi:
end do:
count1-count3;
end proc: # Ridouane Oudra, Feb 21 2023
# second Maple program:
a:= n-> add(`if`(d::odd, d^2*(-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100); # Ridouane Oudra, Feb 21 2023
MATHEMATICA
QP = QPochhammer; s = (1-QP[q]^4*(QP[q^2]^6/QP[q^4]^4))/(4*q) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^2]^4) / 4, {q, 0, n}]; (* Michael Somos, Jun 25 2019 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^2)^(e+1)-1)/(p^2-1), ((-p^2)^(e+1)-1)/(-p^2-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d)))} /* Michael Somos, Aug 09 2006 */
(Haskell)
a002173 n = a050450 n - a050453 n -- Reinhard Zumkeller, Jun 17 2013
(Python)
from math import prod
from sympy import factorint
def A002173(n): return prod(((m:=p**2*(0, 1, 0, -1)[p&3])**(e+1)-1)//(m-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved