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A002173 a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2.
(Formerly M4467 N1895)
10
1, 1, -8, 1, 26, -8, -48, 1, 73, 26, -120, -8, 170, -48, -208, 1, 290, 73, -360, 26, 384, -120, -528, -8, 651, 170, -656, -48, 842, -208, -960, 1, 960, 290, -1248, 73, 1370, -360, -1360, 26, 1682, 384, -1848, -120, 1898, -528, -2208, -8, 2353, 651, -2320, 170 (list; graph; refs; listen; history; text; internal format)
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

N. 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).

F. Jarvis, H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.

J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.

Index entries for sequences mentioned by Glaisher

FORMULA

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

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.: qG'(q)/G(q), with G(q) = Prod[n>=1, (1-q^n)^(4n*A056594(n+1)) ].

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

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-Fran├žois 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 *)

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

CROSSREFS

Equals A050450(n) - A050453(n).

A120030(n) = -4*a(n), if n>0.

Cf. A056594.

Sequence in context: A019432 A211796 A138505 * A050458 A125166 A211785

Adjacent sequences:  A002170 A002171 A002172 * A002174 A002175 A002176

KEYWORD

sign,mult,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)