A002131 -id:A002131 - cp's OEIS frontend

A000593 Sum of odd divisors of n.

1, 1, 4, 1, 6, 4, 8, 1, 13, 6, 12, 4, 14, 8, 24, 1, 18, 13, 20, 6, 32, 12, 24, 4, 31, 14, 40, 8, 30, 24, 32, 1, 48, 18, 48, 13, 38, 20, 56, 6, 42, 32, 44, 12, 78, 24, 48, 4, 57, 31, 72, 14, 54, 40, 72, 8, 80, 30, 60, 24, 62, 32, 104, 1, 84, 48, 68, 18, 96, 48, 72, 13, 74, 38, 124

Offset: 1

N. J. A. Sloane

Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - Michael Somos, May 17 2013
A069289(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A001227 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
For the g.f.s given below by Somos Oct 29 2005, Jovovic, Oct 11 2002 and Arndt, Nov 09 2010, see the Hardy-Wright reference, proof of Theorem 382, p. 312, with x^2 replaced by x. - Wolfdieter Lang, Dec 11 2016
a(n) is also the total number of parts in all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017
It seems that a(n) divides A000203(n) for every n. - Ivan N. Ianakiev, Nov 25 2017 [Yes, see the formula dated Dec 14 2017].
Also, alternating row sums of A126988. - Omar E. Pol, Feb 11 2018
Where a(n) shows the number of equivalence classes of Hurwitz quaternions with norm n (equivalence defined by right multiplication with one of the 24 Hurwitz units as in A055672), A046897(n) seems to give the number of equivalence classes of Lipschitz quaternions with norm n (equivalence defined by right multiplication with one of the 8 Lipschitz units). - R. J. Mathar, Aug 03 2025

			G.f. = x + x^2 + 4*x^3 + x^4 + 6*x^5 + 4*x^6 + 8*x^7 + x^8 + 13*x^9 + 6*x^10 + 12*x^11 + ...

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 496, pp. 69-246, Ellipses, Paris, 2004.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003, p. 312.
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg, 1994, p. 133.
John Riordan, Combinatorial Identities, Wiley, 1968, p. 187.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
Francesca Aicardi, Matricial formulas for partitions, arXiv:0806.1273 [math.NT], 2008.
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999; Canad. J. Math. 51 (1999), 1258-1276.
John A. Ewell, On the sum-of-divisors function, Fib. Q., 45 (2007), 205-207.
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).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Kaya Lakein and Anne Larsen, A Proof of Merca's Conjectures on Sums of Odd Divisor Functions, arXiv:2107.07637 [math.NT], 2021.
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017, also here
Mircea Merca, Congruence identities involving sums of odd divisors function, Proceedings of the Romanian Academy, Series A, Volume 22, Number 2/2021, pp. 119-125.
Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
N. J. A. Sloane, Transforms.
H. J. Stephen Smith, Report on the Theory of Numbers. — Part VI., Report of the 35 Meeting of the British Association for the Advancement of Science (1866). See p. 336.
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Eric Weisstein's World of Mathematics, Partition Function Q.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000203, A000265, A001227, A006128, A050999, A051000, A051001, A051002, A065442, A078471 (partial sums), A069289, A247837 (subset of the primes).

Cf. A301799, A301800.

a000593 = sum . a182469_row  -- Reinhard Zumkeller, May 01 2012, Jul 25 2011

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[j*x^j/(1+x^j): j in [1..2*m]])  )); // G. C. Greubel, Nov 07 2018

[&+[d:d in Divisors(n)|IsOdd(d)]:n in [1..75]]; // Marius A. Burtea, Aug 12 2019

A000593 := proc(n) local d,s; s := 0; for d from 1 by 2 to n do if n mod d = 0 then s := s+d; fi; od; RETURN(s); end;

Table[a := Select[Divisors[n], OddQ[ # ]&]; Sum[a[[i]], {i, 1, Length[a]}], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
f[n_] := Plus @@ Select[ Divisors@ n, OddQ]; Array[f, 75] (* Robert G. Wilson v, Jun 19 2011 *)
a[ n_] := If[ n < 1, 0, Sum[ -(-1)^d n / d, {d, Divisors[ n]}]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# n / # &]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d, {d, Divisors[ n]}]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # < 3, 1, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 15 2015 *)
Array[Total[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] &, {75}] (* Michael De Vlieger, Apr 07 2016 *)
Table[SeriesCoefficient[n Log[QPochhammer[-1, x]], {x, 0, n}], {n, 1, 75}] (* Vladimir Reshetnikov, Nov 21 2016 *)
Table[DivisorSum[n,#&,OddQ[#]&],{n,80}] (* Harvey P. Dale, Jun 19 2021 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d+1) * n/d))}; /* Michael Somos, May 29 2005 */

N=66; x='x+O('x^N); Vec( serconvol( log(prod(j=1,N,1+x^j)), sum(j=1,N,j*x^j)))  /* Joerg Arndt, May 03 2008, edited by M. F. Hasler, Jun 19 2011 */

s=vector(100);for(n=1,100,s[n]=sumdiv(n,d,d*(d%2)));s /* Zak Seidov, Sep 24 2011*/

a(n)=sigma(n>>valuation(n,2)) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import factorint
def A000593(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) # Chai Wah Wu, Sep 09 2021

[sum(k for k in divisors(n) if k % 2) for n in (1..75)] # Giuseppe Coppoletta, Nov 02 2016

Inverse Moebius Transform of [0, 1, 0, 3, 0, 5, ...].
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)).
a(2*n) = A000203(2*n)-2*A000203(n), a(2*n+1) = A000203(2*n+1). - Henry Bottomley, May 16 2000
a(2*n) = A054785(2*n) - A000203(2*n). - Reinhard Zumkeller, Apr 23 2008
Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d divides n} (-1)^(d+1)*n/d, Dirichlet convolution of A062157 with A000027. - Vladeta Jovovic, Sep 06 2002
Sum_{k=1..n} a(k) is asymptotic to c*n^2 where c=Pi^2/24. - Benoit Cloitre, Dec 29 2002
G.f.: Sum_{n>0} n*x^n/(1+x^n). - Vladeta Jovovic, Oct 11 2002
G.f.: (theta_3(q)^4 + theta_2(q)^4 -1)/24.
G.f.: Sum_{k>0} -(-x)^k / (1 - x^k)^2. - Michael Somos, Oct 29 2005
a(n) = A050449(n)+A050452(n); a(A000079(n))=1; a(A005408(n))=A000203(A005408(n)). - Reinhard Zumkeller, Apr 18 2006
From Joerg Arndt, Nov 09 2010: (Start)
G.f.: Sum_{n>=1} (2*n-1) * q^(2*n-1) / (1-q^(2*n-1)).
G.f.: deriv(log(P)) = deriv(P)/P where P = Product_{n>=1} (1 + q^n). (End)
Dirichlet convolution of A000203 with [1,-2,0,0,0,...]. - R. J. Mathar, Jun 28 2011
a(n) = Sum_{k = 1..A001227(n)} A182469(n,k). - Reinhard Zumkeller, May 01 2012
G.f.: -1/Q(0), where Q(k) = (x-1)*(1-x^(2*k+1)) + x*(-1 +x^(k+1))^4/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) = Sum_{k=1..n} k*A000009(k)*A081362(n-k). - Mircea Merca, Feb 26 2014
a(n) = A000203(n) - A146076(n). - Omar E. Pol, Apr 05 2016
a(2*n) = a(n). - Giuseppe Coppoletta, Nov 02 2016
a(n) = n * [x^n] log((-1; x)inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 21 2016
From Wolfdieter Lang, Dec 11 2016: (Start)
G.f.: Sum_{n>=1} x^n*(1+x^(2*n))/(1-x^(2*n))^2, from the second to last equation of the proof to Theorem 382 (with x^2 -> x) of the Hardy-Wright reference, p. 312.
a(n) = Sum_{d|n} (-d)*(-1)^(n/d), commutating factors of the D.g.f. given above by Jovovic, Oct 11 2002. See also the a(n) version given by Jovovic, Sep 06 2002. (End)
a(n) = A000203(n)/A038712(n). - Omar E. Pol, Dec 14 2017
a(n) = A000203(n)/(2^(1 + (A183063(n)/A001227(n))) - 1). - Omar E. Pol, Nov 06 2018
a(n) = A000203(2n) - 2*A000203(n). - Ridouane Oudra, Aug 28 2019
From Peter Bala, Jan 04 2021: (Start)
a(n) = (2/3)*A002131(n) + (1/3)*A002129(n) = (2/3)*A002131(n) + (-1)^(n+1)*(1/3)*A113184(n).
a(n) = A002131(n) - (1/2)*A146076; a(n) = 2*A002131(n) - A000203(n). (End)
a(n) = A000203(A000265(n)) - John Keith, Aug 30 2021
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = A065442 - 1 = 0.60669... . - Amiram Eldar, Dec 14 2024

A002448 Expansion of Jacobi theta function theta_4(x).

Original entry on oeis.org

1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

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

Number 2 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 - 2*q^49 + ...

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.11), p. 6, Eq. (7.324).
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.
J. W. L. Glaisher, On the deduction of series from infinite products, Messenger of Math., 2 (1873), p. 141.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, q-Series Identities
D. Zagier, Elliptic modular forms and their applications in "The 1-2-3 of modular forms", Springer-Verlag, 2008

Cf. A000122, A000203, A010054, A089802.

using Nemo
function JacobiTheta4(len, r)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(r, len, -x)
    [fmpz(coeff(e, j)) for j in 0:len - 1] end
A002448List(len) = JacobiTheta4(len, 1)
A002448List(105) |> println # Peter Luschny, Mar 12 2018

Maple
```
Sum((-x)^(m^2),m=-10..10);
```

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
QP = QPochhammer; s = QP[q]^2/QP[q^2] + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<0, 0, (-1)^n * issquare(n) * 2 - (n==0))}; /* Michael Somos, Jun 17 1999 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A), n))}; /* Michael Somos, May 01 2003 */

from sympy.ntheory.primetest import is_square
def A002448(n): return (-is_square(n) if n&1 else is_square(n))<<1 if n else 1 # Chai Wah Wu, May 17 2023

Expansion of phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q)^2 / eta(q^2) in powers of q. - Michael Somos, May 01 2003
Expansion of 2 * sqrt( k' * K / (2 Pi) ) in powers of q. - Michael Somos, Nov 30 2013
Euler transform of period 2 sequence [ -2, -1, ...]. - Michael Somos, May 01 2003
G.f.: Sum_{k in Z} (-1)^k * x^(k^2) = Product_{k>0} (1 - x^k) / (1 + x^k). - Michael Somos, May 01 2003.
G.f.: 1 - 2 Sum_{k>0} x^k/(1 - x^k) Product_{j=1..k} (1 - x^j) / (1 + x^j). - Michael Somos, Apr 12 2012
a(n) = -2 * b(n) where b(n) is multiplicative and b(2^e) = (-1)^(e/2) if e even, b(p^e) = 1 if p>2 and e even, otherwise 0. - Michael Somos, Jul 07 2006
a(3*n + 1) = -2 * A089802(n), a(9*n) = a(n). - Michael Somos, Jul 07 2006
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A000122(n). a(n) = (-1)^n * A000122(n). a(8*n + 1) = -2 * A010054(n). - Michael Somos, Apr 12 2012
For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1)))*(-1)^(floor(sqrt(n))). - Mikael Aaltonen, Jan 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A010054. - Michael Somos, May 05 2016
a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
G.f.: exp(Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
From Peter Bala, Feb 19 2021: (Start)
G.f: A(q) = eta(q^2)^5 / ( eta(-q)*eta(q^4) )^2.
A(q) = 1 + 2*Sum_{n >= 1} (-1)^n*q^(n*(n+1)/2)/( (1 + q^n) * Product_{k = 1..n} 1 - q^k ).
A(-q)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*q^(2*n-1)/(1 - q^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.
The unsigned sequence has the g.f. 1 + 2*Sum_{n >= 1} q^(n*(n+1)/2) * ( Product_{k = 1..n-1} 1 + q^k ) /( Product_{k = 1..n} 1 + q^(2*k) ) = 1 + 2*q + 2*q^4 + 2*q^9 + .... See Fine, equation 14.43. (End)
Form Peter Bala, Sep 27 2023: (Start)
G.f. A(q) satisfies A(q)*A(-q) = A(q^2)^2.
A(q) = Sum_{n >= 1} (-q)^(n-1)*Product_{k >= n} 1 - q^k. (End)

A008438 Sum of divisors of 2*n + 1.

Original entry on oeis.org

1, 4, 6, 8, 13, 12, 14, 24, 18, 20, 32, 24, 31, 40, 30, 32, 48, 48, 38, 56, 42, 44, 78, 48, 57, 72, 54, 72, 80, 60, 62, 104, 84, 68, 96, 72, 74, 124, 96, 80, 121, 84, 108, 120, 90, 112, 128, 120, 98, 156, 102, 104, 192, 108, 110, 152, 114, 144, 182, 144, 133, 168

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of ways of writing n as the sum of 4 triangular numbers.
Bisection of A000203. - Omar E. Pol, Mar 14 2012
a(n) is also the total number of parts in all partitions of 2*n + 1 into equal parts. - Omar E. Pol, Feb 14 2021

			Divisors of 9 are 1,3,9, so a(4)=1+3+9=13.
F_2(z) = eta(4z)^8/eta(2z)^4 = q + 4q^3 + 6q^5 +8q^7 + 13q^9 + ...
G.f. = 1 + 4*x + 6*x^2 + 8*x^3 + 13*x^4 + 12*x^5 + 14*x^6 + 24*x^7 + 18*x^8 + 20*x^9 + ...
B(q) = q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + 18*q^17 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Ex. (iii).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 19 eq. (6), and p. 283 eq. (8).
W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 12.
H. M. Farkas, I. Kra, Cosines and triangular numbers, Rev. Roumaine Math. Pures Appl., 46 (2001), 37-43.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.31).
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 184, Prop. 4, F(z).
G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ. Press, 1954, page 92 ff.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 10000 terms from T. D. Noe)
H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080)
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
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 3 [Legendre].
H. Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
Michael Somos, Introduction to Ramanujan theta functions
Min Wang and Zhi-Hong Sun, On the number of representations of n as a linear combination of four triangular numbers II, arXiv:1511.00478 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

Cf. A000118, A000593, A005879, A096727, A115607, A129588, A225699/A225700.
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.
Cf. A000203, A002131, A005408, A062731, A099774, A125061, A138741.

a008438 = a000203 . a005408  -- Reinhard Zumkeller, Sep 22 2014

Basis( ModularForms( Gamma0(4), 2), 124) [2]; /* Michael Somos, Jun 12 2014 */

[DivisorSigma(1, 2*n+1): n in [0..70]]; // Vincenzo Librandi, Aug 01 2017

A008438 := proc(n) numtheory[sigma](2*n+1) ; end proc: # R. J. Mathar, Mar 23 2011

DivisorSigma[1, 2 # + 1] & /@ Range[0, 61] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ D[ Series[ Log[ QPochhammer[ -x] / QPochhammer[ x]], {x, 0, 2 n + 1}], x], {x, 0 , 2n}]; (* Michael Somos, Oct 15 2019 *)

PARI
```
{a(n) = if( n<0, 0, sigma( 2*n + 1))};
```

{a(n) = if( n<0, 0, n = 2*n; polcoeff( sum( k=1, (sqrtint( 4*n + 1) + 1)\2, x^(k^2 - k), x * O(x^n))^4, n))}; /* Michael Somos, Sep 17 2004 */

{a(n) = my(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 / eta(x^2 + A))^4, n))}; /* Michael Somos, Sep 17 2004 */

ModularForms( Gamma0(4), 2, prec=124).1;  # Michael Somos, Jun 12 2014

Expansion of q^(-1/2) * (eta(q^2)^2 / eta(q))^4 = psi(q)^4 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Apr 11 2004
Expansion of Jacobi theta_2(q)^4 / (16*q) in powers of q^2. - Michael Somos, Apr 11 2004
Euler transform of period 2 sequence [4, -4, 4, -4, ...]. - Michael Somos, Apr 11 2004
a(n) = b(2*n + 1) where b() is multiplicative and b(2^e) = 0^n, b(p^e) =(p^(e+1) - 1) / (p - 1) if p>2. - Michael Somos, Jul 07 2004
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 8*w*v^2 + 16*w^2*v - u^2*w - Michael Somos, Apr 08 2005
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^3), B(q^9)) where f(u, v, w) = v^4 - 30*u*v^2*w + 12*u*v*w*(u + 9*w) - u*w*(u^2 + 9*w*u + 81*w^2).
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2^3 + u1^2*u6 + 3*u2*u3^2 + 27*u6^3 - u1*u2*u3 - 3*u1*u3*u6 - 7*u2^2*u6 - 21*u2*u6^2. - Michael Somos, May 30 2005
G.f.: Sum_{k>=0} (2k + 1) * x^k / (1 - x^(2k + 1)).
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004
G.f. Sum_{k>=0} a(k) * x^(2*k + 1) = x * (Product_{k>0} (1 - x^(4*k))^2 / (1 - x^(2*k)))^4 = x * (Sum_{k>0} x^(k^2 - k))^4 = Sum_{k>0} k * (x^k / (1 - x^k) - 3 * x^(2*k) / (1 - x^(2*k)) + 2 * x^(4*k) / (1 - x^(4*k))). - Michael Somos, Jul 07 2004
Number of solutions of 2*n + 1 = (x^2 + y^2 + z^2 + w^2) / 4 in positive odd integers. - Michael Somos, Apr 11 2004
8 * a(n) = A005879(n) = A000118(2*n + 1). 16 * a(n) = A129588(n). a(n) = A000593(2*n + 1) = A115607(2*n + 1).
a(n) = A000203(2*n+1). - Omar E. Pol, Mar 14 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A096727. Michael Somos, Jun 12 2014
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 4*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017
From Peter Bala, Jan 10 2021: (Start)
a(n) = A002131(2*n+1).
G.f.: Sum_{n >= 0} x^n*(1 + x^(2*n+1))/(1 - x^(2*n+1))^2. (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 8. - Vaclav Kotesovec, Aug 07 2022
Convolution of A125061 and A138741. - Michael Somos, Mar 04 2023

Comments from Len Smiley, Enoch Haga

A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 1, 3, 7, 14, 28, 58, 106, 201, 372, 669, 1187, 2101, 3624, 6229, 10591, 17796, 29659, 49107, 80492, 131157, 212237, 341084, 544883, 865717, 1367233, 2148552, 3359490, 5227270, 8096544, 12486800, 19174319, 29326306, 44678825, 67811375, 102549673, 154545549

Offset: 0

Joerg Arndt, Jun 24 2011

nonn

Euler transform of A002131. - Vaclav Kotesovec, Mar 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A061256 (1/Product_{k>=1} P(x^k)^k where P(x) = Product_{k>=1} (1 - x^k)).

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, Jul 03 2017 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

N=66;  x='x+O('x^N);
Q(x)=prod(k=1,N,1+x^k);
gf=prod(k=1,N, Q(x^k)^k );
Vec(gf) /* Joerg Arndt, Jun 24 2011 */

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 09 2017

a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018

A001934 Expansion of 1/theta_4(q)^2 in powers of q.

Original entry on oeis.org

1, 4, 12, 32, 76, 168, 352, 704, 1356, 2532, 4600, 8160, 14176, 24168, 40512, 66880, 108876, 174984, 277932, 436640, 679032, 1046016, 1597088, 2418240, 3632992, 5417708, 8022840, 11802176, 17252928, 25070568, 36223424, 52053760, 74414412

Offset: 0

N. J. A. Sloane, Simon Plouffe

Euler transform of period 2 sequence [ 4, 2, ...].
The Cayley reference actually is to A004403. - Michael Somos, Feb 24 2011
Number of overpartition pairs, see Lovejoy reference. - _Joerg Arndt, Apr 03 2011
In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^m and m>=1, then a(n) ~ exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)). - Vaclav Kotesovec, Aug 17 2015

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
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).

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
B. Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
Jeremy Lovejoy, Overpartition pairs, Annales de l'institut Fourier, vol.56, no.3, p.781-794, 2006.

Cf. A000122, A000203, A004403, A002318, A002513, A015128, A004404-A004425.

# JacobiTheta4 is defined in A002448.
A001934List(len) = JacobiTheta4(len, -2)
A001934List(33) |> println # Peter Luschny, Mar 12 2018

Maple
```
mul((1+x^n)^2/(1-x^n)^2,n=1..256);
```

CoefficientList[Series[1/EllipticTheta[4, 0, q]^2, {q, 0, 32}], q]  (* Jean-François Alcover, Jul 18 2011 *)
nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
QP = QPochhammer; s = QP[q^2]^2/QP[q]^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

my(N=33, x='x+O('x^N)); Vec(prod(i=1, N, (1+x^i)^2/(1-x^i)^2))

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4, n))} /* Michael Somos, Feb 09 2006 */

G.f.: Product ( 1 - x^k )^{-c(k)}, c(k) = 4, 2, 4, 2, 4, 2, ....
G.f.: Product{i>=1} (1+x^i)^2/(1-x^i)^2. - Jon Perry, Apr 04 2004
Expansion of eta(q^2)^2/eta(q)^4 in powers of q, where eta(x)=prod(n>=1,1-q^n).
a(n) = (-1)^n * A004403(n). a(n) = 4 * A002318(n) unless n=0. - Michael Somos, Feb 24 2011
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(15/4) * n^(5/4)) * (1 - 15/(8*Pi*sqrt(2*n)) + 105/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 17 2015, extended Jan 22 2017
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
The g.f. A(q^2) = 1/(F(q)*F(-q)), where F(q) = theta_3(q) = Sum_{n = -oo..oo} q^(n^2) is the g.f. of A000122. Cf. A002513. - Peter Bala, Sep 26 2023

More terms from James Sellers, Sep 08 2000

Edited by N. J. A. Sloane, May 13 2008 to remove an incorrect g.f.

A186690 Expansion of - (1/8) theta_3''(0, q) / theta_3(0, q) in powers of q.

Original entry on oeis.org

1, -2, 4, -4, 6, -8, 8, -8, 13, -12, 12, -16, 14, -16, 24, -16, 18, -26, 20, -24, 32, -24, 24, -32, 31, -28, 40, -32, 30, -48, 32, -32, 48, -36, 48, -52, 38, -40, 56, -48, 42, -64, 44, -48, 78, -48, 48, -64, 57, -62, 72, -56, 54, -80, 72, -64, 80, -60, 60, -96, 62, -64

Offset: 1

Michael Somos, Feb 25 2011

If A(x) is the generating function then 1 / Pi = 8 A( exp( -Pi) ). [Plouffe, equation 1.2]

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

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

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Equation (5.1.29.8).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Simon Plouffe, Identities inspired by the Ramanujan Notebooks, Second series, arXiv:1101.6066 [math.NT], 2011.
R. Sivaraman, José Luis López-Bonilla, and Sergio Vidal-Beltrán, On the Polynomial Structure of r_k(n), Indian J. Adv. Math. (2023) Vol. 3, Iss. 2, A1162044124.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002131, A186111.

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1/8) (EllipticE[m] - (1 - m) EllipticK[m]) EllipticK[m]/(Pi/2)^2, {q, 0, n}]];

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

from math import prod
from sympy import factorint
def A186690(n): return (1 if n&1 else -1)*prod((p**(e+1)-1)//(p-1) if p&1 else 1<Chai Wah Wu, Jun 23 2024

Multiplicative with a(2^e) = -(2^e) if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2.
Expansion of (E - (1 - k^2) * K) * K / (2 Pi^2) in powers of the nome q where K, E are complete elliptic integrals.
Expansion of (1/2) x (d phi(x) / dx) / phi(x) in powers of x where phi() is a Ramanujan theta function.
G.f.: Sum_{k>0} - (-1)^k * k * x^k / (1 - x^(2*k)) = Sum_{k>0} x^(2*k-1) / (1 + x^(2*k-1))^2 = (Sum_{k>0} n^2 x^(n^2)) / (Sum_k x^(n^2)).
Dirichlet g.f. zeta(s) *zeta(s-1) *(1-7*2^(-s)+14*4^(-s)-8^(1-s)) / (1-2^(1-s)). - R. J. Mathar, Jun 01 2011
a(n) = -(-1)^n * A002131(n).
MOBIUS transform is A186111. - Michael Somos, Apr 25 2015

A054785 a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.

Original entry on oeis.org

2, 4, 8, 8, 12, 16, 16, 16, 26, 24, 24, 32, 28, 32, 48, 32, 36, 52, 40, 48, 64, 48, 48, 64, 62, 56, 80, 64, 60, 96, 64, 64, 96, 72, 96, 104, 76, 80, 112, 96, 84, 128, 88, 96, 156, 96, 96, 128, 114, 124, 144, 112, 108, 160, 144, 128, 160, 120, 120, 192, 124, 128, 208

Offset: 1

Labos Elemer, May 22 2000

Sum of divisors of 2*n that do not divide n. - Franklin T. Adams-Watters, Oct 04 2018

a(n) = 2*n iff n = 2^k, k >= 0 (A000079). - Bernard Schott, Mar 24 2020

			n=9: sigma(18)=18+9+6+3+2+1=39, sigma(9)=9+3+1=13, a(9)=39-13=26.

Paul Tek, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Wang-Sun formula in GL(Z/2kZ), Integers, Vol. 23 (2023), #A37.

Cf. A000079, A000203, A000593, A111003, A002131, A320059.

[DivisorSigma(1, 2*n) - DivisorSigma(1, n): n in [1..70]]; Vincenzo Librandi, Oct 05 2018

a:= proc(n) local e;
  e:= 2^padic:-ordp(n,2);
  2*e*numtheory:-sigma(n/e)
end proc:
map(a, [$1..100]); # Robert Israel, Jul 05 2016

Table[DivisorSigma[1,2n]-DivisorSigma[1,n],{n,70}] (* Harvey P. Dale, May 11 2014 *)
Table[CoefficientList[Series[-Log[EllipticTheta[4, 0, x]], {x, 0, 80}],x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)

a(n)=sigma(2*n)-sigma(n) \\ Charles R Greathouse IV, Feb 13 2013

a(n) = A000203(2n) - A000203(n).
a(n) = 2*A002131(n).
a(2*n) = A000203(n) + A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
L.g.f.: -log(EllipticTheta(4,0,x)) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
G.f.: Sum_{k>=1} 2*k*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Oct 23 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). - Amiram Eldar, Jan 19 2024

cp's OEIS Frontend

A164557 Numbers k such that s(k) = s(k+1), where s(k) is the sum of divisors d of k such that k/d is odd (A002131).

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A301798 Expansion of Product_{k>=1} (1 + x^k)^A002131(k).

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A350146 Partial sums of A002131.

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A000593 Sum of odd divisors of n.

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A002448 Expansion of Jacobi theta function theta_4(x).

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A008438 Sum of divisors of 2*n + 1.

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