A002173 -id:A002173 - cp's OEIS frontend

A000141 Number of ways of writing n as a sum of 6 squares.

1, 12, 60, 160, 252, 312, 544, 960, 1020, 876, 1560, 2400, 2080, 2040, 3264, 4160, 4092, 3480, 4380, 7200, 6552, 4608, 8160, 10560, 8224, 7812, 10200, 13120, 12480, 10104, 14144, 19200, 16380, 11520, 17400, 24960, 18396, 16440, 24480, 27200

Offset: 0

N. J. A. Sloane

The relevant identity for the o.g.f. is theta_3(x)^6 = 1 + 16*Sum_{j>=1} j^2*x^j/(1 + x^(2*j)) - 4*Sum_{j >=0} (-1)^j*(2*j+1)^2 *x^(2*j+1)/(1 - x^(2*j+1)), See the Hardy-Wright reference, p. 315, first equation. - Wolfdieter Lang, Dec 08 2016

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124
S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, 2013, Preprint submitted to Journal of Geometry and Physics.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Index entries for sequences related to sums of squares

Row d=6 of A122141 and of A319574, 6th column of A286815.

Cf. A050470, A002173.

a000141 0 = 1
a000141 n = 16 * a050470 n - 4 * a002173 n
-- Reinhard Zumkeller, Jun 17 2013

(sum(x^(m^2),m=-10..10))^6;
# Alternative:
A000141list := proc(len) series(JacobiTheta3(0, x)^6, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000141list(40); # Peter Luschny, Oct 02 2018

Table[SquaresR[6, n], {n, 0, 40}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[6,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)
EllipticTheta[3, 0, z]^6 + O[z]^40 // CoefficientList[#, z]& (* Jean-François Alcover, Dec 05 2019 *)

from math import prod
from sympy import factorint
def A000141(n):
    if n == 0: return 1
    f = [(p,e,(0,1,0,-1)[p&3]) for p,e in factorint(n).items()]
    return (prod((p**(e+1<<1)-c)//(p**2-c) for p, e, c in f)<<2)-prod(((k:=p**2*c)**(e+1)-1)//(k-1) for p, e, c in f)<<2 # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*6)
Q.representation_number_list(40) # Peter Luschny, Jun 20 2014

Expansion of theta_3(z)^6.
a(n) = 4( Sum_{ d|n, d ==3 mod 4} d^2 - Sum_{ d|n, d ==1 mod 4} d^2 ) + 16( Sum_{ d|n, n/d ==1 mod 4} d^2 - Sum_{ d|n, n/d ==3 mod 4} d^2 ) [Jacobi]. [corrected by Sean A. Irvine, Oct 01 2009]
a(n) = 16*A050470(n) - 4*A002173(n). - Michel Marcus, Dec 15 2012
a(n) = (12/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A008440 Expansion of Jacobi theta constant theta_2^6 /(64q^(3/2)).

Original entry on oeis.org

1, 6, 15, 26, 45, 66, 82, 120, 156, 170, 231, 276, 290, 390, 435, 438, 561, 630, 651, 780, 861, 842, 1020, 1170, 1095, 1326, 1431, 1370, 1716, 1740, 1682, 2016, 2145, 2132, 2415, 2550, 2353, 2850, 3120, 2810, 3321, 3486, 3285, 3906, 4005, 3722, 4350

Offset: 0

N. J. A. Sloane

Number of representations of n as sum of 6 triangular numbers. - Michel Marcus, Oct 24 2012. See the Ono et al. link.

			G.f. = 1 + 6*x + 15*x^2 + 26*x^3 + 45*x^4 + 66*x^5 + 82*x^6 + ... - _Michael Somos_, Jun 25 2019
G.f. = q^3 + 6*q^7 + 15*q^11 + 26*q^15 + 45*q^19 + 66*q^23 + 82*q^27 + ...

B. C. Berndt, Fragments by Ramanujan on Lambert series, in Number Theory and Its Applications, K. Gyory and S. Kanemitsu, eds., Kluwer, Dordrecht, 1999, pp. 35-49, see Entry 6.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
B. C. Berndt, Fragments by Ramanujan on Lambert series.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 4.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809, A002173.

CoefficientList[(QPochhammer[q^2]^2 / QPochhammer[q])^6 + O[q]^50, q] (* Jean-François Alcover, Nov 05 2015 *)
a[ n_] := If[ n < 0, 0, -DivisorSum[ 4 n + 3, Re[I^(# - 1)] #^2 &] / 8]; (* Michael Somos, Jun 25 2019 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x * O(x^n))^6, n))}; /* Michael Somos, May 23 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^6, n))}; /* Michael Somos, May 23 2006 */

{a(n)= -sumdiv(4*n + 3, d, real(I^(d-1))*d^2)/8}; /* Michael Somos, Oct 24 2012 */

Expansion of Ramanujan phi^6(q) in powers of q.
Expansion of q^(-3/4)(eta(q^2)^2/eta(q))^6 in powers of q.
Euler transform of period 2 sequence [6, -6, ...]. - Michael Somos, May 23 2006
G.f.: (Sum_{n>=0} x^((n^2+n)/2))^6.
a(n) = (-1/8)*Sum_{d divides (4n+3)} Chi_2(4;d)*d^2. - Michel Marcus, Oct 24 2012. See the Ono et al. link. Theorem 4.
a(n) =(-1/8)*A002173(4*n+3). This is the preceding formula. - Wolfdieter Lang, Jan 12 2017
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 6*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A322143 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, -2, 1, 1, 1, -8, 1, 2, 1, 1, -26, 1, 6, 0, 1, 1, -80, 1, 26, -2, 0, 1, 1, -242, 1, 126, -8, -6, 1, 1, 1, -728, 1, 626, -26, -48, 1, 1, 1, 1, -2186, 1, 3126, -80, -342, 1, 7, 2, 1, 1, -6560, 1, 15626, -242, -2400, 1, 73, 6, 0, 1, 1, -19682, 1, 78126, -728, -16806, 1, 703, 26, -10, 0

Offset: 1

Ilya Gutkovskiy, Nov 28 2018

			Square array begins:
  1,  1,   1,    1,    1,     1,  ...
  1,  1,   1,    1,    1,     1,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...
  1,  1,   1,    1,    1,     1,  ...
  2,  6,  26,  126,  626,  3126,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...

Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.

Cf. A109974, A322081, A322082, A322083, A322084.

Table[Function[k, SeriesCoefficient[Sum[(-1)^(j - 1) (2 j - 1)^k x^(2 j - 1)/(1 - x^(2 j - 1)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} (-1)^(j-1)*(2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

Original entry on oeis.org

1, -4, -4, 32, -4, -104, 32, 192, -4, -292, -104, 480, 32, -680, 192, 832, -4, -1160, -292, 1440, -104, -1536, 480, 2112, 32, -2604, -680, 2624, 192, -3368, 832, 3840, -4, -3840, -1160, 4992, -292, -5480, 1440, 5440, -104, -6728, -1536, 7392, 480, -7592, 2112, 8832, 32, -9412, -2604

Offset: 0

Michael Somos, Jun 05 2006

sign

Number 8 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).

			G.f. = 1 - 4*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

G. C. Greubel, Table of n, a(n) for n = 0..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Frazer Jarvis and Helena A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.
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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000141, A002173, A050470, A128692.

A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* Michael Somos, May 24 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 24 2015 *)

{a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d^2 * kronecker( -4, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^6 / eta(x^4 + A)^4, n))};

Expansion of eta(q)^4 * eta(q^2)^6 / eta(q^4)^4 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).
Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].
G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).
Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.
a(n) = -4 * A002173(n) unless n=0.
Convolution of A000141 and A128692.

A103440 a(n) = Sum[d|n, d==1 (mod 3), d^2] - Sum[d|n, d==2 (mod 3), d^2].

Original entry on oeis.org

1, -3, 1, 13, -24, -3, 50, -51, 1, 72, -120, 13, 170, -150, -24, 205, -288, -3, 362, -312, 50, 360, -528, -51, 601, -510, 1, 650, -840, 72, 962, -819, -120, 864, -1200, 13, 1370, -1086, 170, 1224, -1680, -150, 1850, -1560, -24, 1584, -2208, 205, 2451, -1803, -288, 2210, -2808, -3, 2880, -2550

Offset: 1

Ralf Stephan, Feb 11 2005

			G.f. = q - 3*q^2 + q^3 + 13*q^4 - 24*q^5 - 3*q^6 + 50*q^7 - 51*q^8 + q^9 + ...

Robert Israel, Table of n, a(n) for n = 1..10000
G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.

Equals A103637(n) - A103638(n). Cf. A002173.

A109041(n) = -9 * a(n) unless n=0. A014985(n) = a(2^n). -24 * A134340(n) = a(6*n+5).

f:= proc(n) local D,d;
  D:= numtheory:-divisors(n/3^padic:-ordp(n,3));
  -add((-1)^(d mod 3)*d^2, d = D)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 16 2018

a[n_] := Sum[m=Mod[d, 3]; (Boole[m==1]-Boole[m==2]) d^2, {d, Divisors[n]}];
Array[a, 56] (* Jean-François Alcover, Aug 16 2018 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ x]^9 / QPochhammer[ x^3]^3) / 9, {x, 0, n}]; (* Michael Somos, Sep 07 2018 *)

{a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( -3, d)))}; /* Michael Somos, Oct 21 2007 */

{a(n) = my(A, p, e, a0, a1, x, y, z); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, z = kronecker( -3, p) * p^2; a0 = 1; a1 = y = z + 1; for(i=2, e, x = y * a1 - z * a0; a0 = a1; a1 = x); a1)))}; /* Michael Somos, Oct 21 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^9 / eta(x^3 + A)^3) / 9, n))}; /* Michael Somos, Oct 21 2007 */

G.f.: F(q) = Sum_{n>=1} A049347(n-1) * n^2 * q^n / (1 - q^n).
G.f.: F(q) = -q * G'(q) / (9*G(q)), with G(q) = Product_{n>=1} (1 - q^n)^(9*n * A049347(n-1)).
a(n) is multiplicative with a(3^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - z * a(p^(e-2)) where z = Kronecker(-3, p) * p^2 and a(p) = z + 1.
a(3*n) = a(n).
G.f.: Sum_{k>0} x^k * (1 - x^k - 6*x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3. - Michael Somos, Oct 21 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v + w + 3*v^2 - 8*w^2 + 6*v*w - 8*u*w + 6*u*v - 9*v^3 - 54*u*v*w + 72*u*w^2 - 9*u^2*w. - Michael Somos, Dec 23 2007

A050458 Difference between Sum_{d|n, d == 1 (mod 4)} d^2 and Sum_{d|n, d == 3 (mod 4)} d^2.

Original entry on oeis.org

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, 2810, 656, 3120, 48, 2880, 842

Offset: 1

Vladeta Jovovic, Feb 15 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

a(n) = |A002173(n)|.

f[p_, e_] := If[Mod[p, 4] == 1, ((p^2)^(e+1)-1)/(p^2-1), ((p^2)^(e+1)+(-1)^e)/(p^2+1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)

{a(n)=if(n<1, 0, abs(sumdiv(n, d, d^2*kronecker(-4,d))))} /* Michael Somos, Aug 09 2006 */

{a(n)= local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(p%4==1, ((p^2)^(e+1)-1)/(p^2-1), ((p^2)^(e+1)+(-1)^e)/(p^2+1)))))) } /* Michael Somos, Aug 09 2006 */

{a(n)=if(n<1, 0, polcoeff( sum(k=1,n, x^k/(1+x^(2*k))*(k/2^valuation(k,2))^2, x*O(x^n)), n))} /* Michael Somos, Aug 09 2006 */

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)^e)/(p^2+1) if p == 3 (mod 4). - Michael Somos, Aug 09 2006

Edited by Michael Somos, Aug 09 2006

A050459 a(n) = Sum_{d|n, d==1 mod 4} d^3 - Sum_{d|n, d==3 mod 4} d^3.

Original entry on oeis.org

1, 1, -26, 1, 126, -26, -342, 1, 703, 126, -1330, -26, 2198, -342, -3276, 1, 4914, 703, -6858, 126, 8892, -1330, -12166, -26, 15751, 2198, -18980, -342, 24390, -3276, -29790, 1, 34580, 4914, -43092, 703, 50654, -6858, -57148, 126, 68922, 8892, -79506, -1330

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative because it is the Inverse Möbius transform of [1 0 -3^3 0 5^3 0 -7^3 ...], which is multiplicative. - Christian G. Bower, May 18 2005

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).
Index entries for sequences mentioned by Glaisher.

Column k=3 of A322143.
Glaisher's E_i (i=0..12): A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.
Cf. A050451, A050454.

A050459 := proc(n) local a; a := 0 ; for d in numtheory[divisors](n) do if d mod 4 = 1 then a := a+d^3 ; elif d mod 4 = 3 then a := a-d^3 ; end if; end do;  a ; end proc:
seq(A050459(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

s[n_, r_] := DivisorSum[n, #^3 &, Mod[#, 4]==r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Dec 06 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^3)^(e+1)-1)/(p^3-1), ((-p^3)^(e+1)-1)/(-p^3-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

a(n) = A050451(n) - A050454(n).
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^3*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^3)^(e+1)-1)/(p^3-1) if p == 1 (mod 4) and ((-p^3)^(e+1)-1)/(-p^3-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^3*sin(d*Pi/2). - Ridouane Oudra, Jun 02 2024

A138502 Expansion of q^(-1/2) * (eta(q)^4 * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

Original entry on oeis.org

1, -8, 26, -48, 73, -120, 170, -208, 290, -360, 384, -528, 651, -656, 842, -960, 960, -1248, 1370, -1360, 1682, -1848, 1898, -2208, 2353, -2320, 2810, -3120, 2880, -3480, 3722, -3504, 4420, -4488, 4224, -5040, 5330, -5208, 5760, -6240, 5905, -6888, 7540, -6736, 7922, -8160, 7680

Offset: 0

Michael Somos, Mar 20 2008

sign

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

			G.f. = 1 - 8*x + 26*x^2 - 48*x^3 + 73*x^4 - 120*x^5 + 170*x^6 - 208*x^7 + ...
g.f. = q - 8*q^3 + 26*q^5 - 48*q^7 + 73*q^9 - 120*q^11 + 170*q^13 - 208*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002173, A122854.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^4 QPochhammer[ q^4]^2 / QPochhammer[ q^2]^3)^2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := If[ n < 0, 0, Times @@ (Function[ {p, e}, If[ p < 3, 2 - p, With[{f = (-1)^Quotient[p, 2]}, f ((f p^2)^(e + 1) - 1)/(p^2 - f)]]]) @@@ FactorInteger[2 n + 1]]; (* Michael Somos, Aug 26 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, d^2 * kronecker( -4, d)))};

{a(n) = my(A, p, e, f); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, f = (-1)^(p\2); f * ((f*p^2)^(e+1) - 1) / (p^2 - f))))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n) ; polcoeff( (eta(x + A)^4 * eta(x^4 + A)^2 / eta(x^2 + A)^3)^2, n))}

Expansion of (phi(-q)^2 * psi(q^2))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -8, -2, -8, -6, ...].
a(n) = b(2*n + 1) where b() is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), b(p^e) = (-(-p^2)^(e+1) + 1) / (p^2 + 1) if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 256 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138501.
G.f.: (Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^2 / (1 + x^k))^2.
a(n) = (-1)^n * A122854(n) = A002173(2*n + 1).

A138505 Expansion of ((phi(q) * phi(-q^2)^2)^2 - 1) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Mar 21 2008

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

			G.f. = q - q^2 - 8*q^3 - q^4 + 26*q^5 + 8*q^6 - 48*q^7 - q^8 + 73*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002173, A138504.

a[ n_] := If[ n < 1, 0, - DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] (-1)^(n/#) &]]; (* Michael Somos, Sep 25 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d) * -(-1)^(n/d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4))^2 - 1) / 4, n))};

a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), a(p^e) = ((-p^2)^(e+1) - 1) / ( -p^2 - 1) if p == 3 (mod 4).
G.f.: Sum_{k>0} -(-1)^k * (2*k-1)^2 * x^(2*k-1) / (1 + x^(2*k-1)).
a(2*n) = (-1)^n * a(n).
4 * a(n) = A138504(n) unless n=0.
a(n) = -(-1)^n * A002173(n). - Michael Somos, Sep 25 2015

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