A097723 -id:A097723 - cp's OEIS frontend

A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).

1, 1, 1, 2, 0, 1, 2, 1, 1, 1, 1, 0, 3, 1, 0, 2, 1, 1, 1, 0, 1, 3, 1, 2, 0, 0, 1, 2, 1, 0, 3, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 3, 0, 1, 3, 0, 0, 2, 1, 0, 0, 1, 2, 4, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 1, 2, 1, 0, 3, 0, 1, 4, 0, 1, 0, 1, 0, 2, 1, 1, 2, 0, 0, 2, 2, 1, 3, 0, 0, 2, 2, 1, 0, 2, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of representations of n as the sum of a triangular number and twice a triangular number. - James Sellers, Dec 21 2005
Also the number of positive odd solutions to equation x^2 + 2*y^2 = 8*n + 3. - Seiichi Manyama, May 28 2017

			G.f. = 1 + x + x^2 + 2*x^3 + x^5 + 2*x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^12 + ...
G.f. = q^3 + q^11 + q^19 + 2*q^27 + q^43 + 2*q^51 + q^59 + q^67 + q^75 + q^83 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
Iana I. Anguelova, The two bosonizations of the CKP hierarchy: overview and character identities, arXiv:1708.04992 [math-ph], 2017.
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A027414, A097723, A033761-A033807, A082564.

A := Basis( ModularForms( Gamma1(32), 1), 840); A[4] + A[12]; /* Michael Somos, Jan 31 2015 */

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A002325 := proc(n) sigmamr(n,8,1)+sigmamr(n,8,3)-sigmamr(n,8,5)-sigmamr(n,8,7) ; end proc:
A033761 := proc(n) A002325(8*n+3)/2 ; end proc:
seq(A033761(n),n=0..90) ; # R. J. Mathar, Mar 23 2011

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] / 4, {q, 0, 2 n + 3/4}]; (* Michael Somos, Nov 16 2011 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^4]^2/QP[q]) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A), n))}; /* Michael Somos, Jul 05 2006 */

Euler transform of period 4 sequence [1, 0, 1, -2, ...]. - Vladeta Jovovic, Sep 14 2004
Expansion of psi(q) * psi(q^2) in powers of q where psi() is a Ramanujan theta function.
Expansion of q^(-3/8) * eta(q^2) * eta^2(q^4) / eta(q) in powers of q. - Michael Somos, Jul 05 2006
Expansion of q^(-3/4) * (theta_2(q) * theta_2(q^2)) / 4 in powers of q^2. - Michael Somos, Jul 05 2006
Given g.f. A(x), then B(x) = x^3 * A(x^8) satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^2 + 3*u2^2*u3^4 - 4*u1*u2*u3*u6 * (u2^2 + 3*u6^2). - Michael Somos, Jul 05 2006
a(n) = A002325(8*n+3)/2. [Hirschhorn] - R. J. Mathar, Mar 23 2011
a(n) = A027414(8*n + 3). - Michael Somos, Nov 16 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082564. - Michael Somos, Jan 31 2015
From Peter Bala, Jan 07 2021: (Start)
G.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(8*n + 3)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(8*n + 3)). Cf. A121444. (End)
A(q^2) = (1/2)*Sum_{k >= 0} q^k/(1 + q^(4*k+3)) + (1/2)*Sum_{k >= 0} q^(3*k)/(1 + q^(4*k+1)) - set z = 1 and replace q with q^2 in Anguelova, equation 3.35. - Peter Bala, Mar 03 2021

More terms from Vladeta Jovovic, Sep 14 2004

A097057 Number of integer solutions to a^2 + b^2 + 2c^2 + 2d^2 = n.

Original entry on oeis.org

1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240

Offset: 0

N. J. A. Sloane, Sep 15 2004

a^2 + b^2 + 2*c^2 + 2*d^2 is another (cf. A000118) of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008

			1 + 4*q + 8*q^2 + 16*q^3 + 24*q^4 + 24*q^5 + 32*q^6 + 32*q^7 + 24*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 31.
Jesse Ira Deutsch, Bumby's technique and a result of Liouville on a quadratic form, Integers 8 (2008), no. 2, A2, 20 pp. MR2438287 (2009g:11047).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).
S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917), 11-21).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Jesse Ira Deutsch, A quaternionic proof of the representation formula of a quaternary quadratic form, J. Number Theory 113 (2005), no. 1, 149-174. MR2141762 (2006b:11033).
Y. Mimura, Almost Universal Quadratic Forms.
Olivia X. M. Yao and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1-9. MR3141622.

a^2 + b^2 + 2*c^2 + m*d^2 = n: this sequence (m=2), A320124 (m=3), A320125 (m=4), A320126 (m=5), A320127 (m=6), A320128 (m=7), A320130 (m=8), A320131 (m=9), A320132 (m=10), A320133 (m=11), A320134 (m=12), A320135 (m=13), A320136 (m=14).

Cf. A000118, A004011, A008438, A097723, A112610.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}] (* Michael Somos, Jul 05 2011 *)
f[p_, e_] := (p^(e+1)-1)/(p-1); f[2, 1] = 2; f[2, e_] := 6; a[0] = 1; a[1] = 4; a[n_] := 4 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Aug 22 2023 *)

{a(n) = local(t); if( n<1, n>=0, t = 2^valuation( n, 2); 4 * sigma(n/t) * if( t>2, 6, t))} \\ Michael Somos, Sep 17 2004

{a(n) = local(A = x * O(x^n)); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^6 / (eta(x + A) * eta(x^8 + A))^4, n)} \\ Michael Somos, Sep 17 2004

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], n)[n])} \\ Michael Somos, Oct 29 2005

A097057(n)=if(n,sigma(n>>n=valuation(n,2))*if(n>1,24,4<M. F. Hasler, May 07 2018

Euler transform of period 8 sequence [4, -2, 4, -8, 4, -2, 4, -4, ...]. - Michael Somos, Sep 17 2004
Multiplicative with a(n) = 4*b(n), b(2) = 2, b(2^e) = 6 if e > 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2. - Michael Somos, Sep 17 2004
Expansion of (eta(q^2) * eta(q^4))^6 / (eta(q) * eta(q^8))^4 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 05 2011
G.f.: (theta_3(q) * theta_3(q^2))^2.
G.f.: Product_{k>0} ((1-x^(2k))(1-x^(4k)))^6/((1-x^k)(1-x^(8k)))^4.
G.f.: 1 + Sum_{k>0} 8 * x^(4*k) / (1 + x^(4*k))^2 + 4 * x^(2*k-1) / (1 - x^(2*k-1))^2 = 1 + Sum_{k>0} (2 + (-1)^k) * 4*k * x^(2*k) / (1 + x^(2*k)) + 4*(2*k - 1) * x^(2*k-1) / (1 - x^(2*k - 1)). - Michael Somos, Oct 22 2005
a(2*n) = A000118(n). a(2*n + 1) = 4 * A008438(n). a(4*n) = A004011(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 2) = 8 * A008438(n). a(4*n + 3) = 16 * A097723(n). - Michael Somos, Jul 05 2011

Added keyword mult and minor edits by M. F. Hasler, May 07 2018

A098098 a(n) = sigma(6*n+5)/6.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72

Offset: 0

Vladeta Jovovic, Sep 14 2004

Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020

			G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...

Ivan Neretin, Table of n, a(n) for n = 0..10000
Nayandeep Deka Baruah and Kallol Nath, Infinite families of arithmetic identities and congruences for bipartitions with 3-cores, Journal of Number Theory, Volume 149, April 2015, Pages 92-104.

Cf. A000203, A016969, A033686, A097723, A121443, A125514, A133739, A134077, A134078, A134079, A186100, A232343, A232356, A252650, A238831.

Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */

Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)

PARI
```
a(n) = sigma(6*n + 5)/6
```

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) =  A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
From Amiram Eldar, Dec 16 2022: (Start)
a(n) = A000203(A016969(n))/6.
Sum_{k=1..n} a(k) = (Pi^2/18) * n^2 + O(n*log(n)). (End)

A033686 One-ninth of theta series of A2[hole]^2.

Original entry on oeis.org

1, 2, 5, 4, 8, 6, 14, 8, 14, 10, 21, 16, 20, 14, 28, 16, 31, 18, 40, 20, 32, 28, 42, 24, 38, 32, 62, 28, 44, 30, 56, 40, 57, 34, 70, 36, 72, 38, 70, 48, 62, 52, 85, 44, 68, 46, 112, 56, 74, 50, 100, 64, 80, 64, 98, 56, 108, 58, 124, 60, 112, 76, 112, 64, 98, 66, 155, 80, 104

Offset: 0

N. J. A. Sloane

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Number of partition pairs of n where each partition is 3-core (see Theorem 2.1 of Wang link). - Michel Marcus, Jul 14 2015

			G.f. = 1 + 2*x + 5*x^2 + 4*x^3 + 8*x^4 + 6*x^5 + 14*x^6 + 8*x^7 + 14*x^8 + ...
G.f. = q^2 + 2*q^5 + 5*q^8 + 4*q^11 + 8*q^14 + 6*q^17 + 14*q^20 + 8*q^23 + ...
Theta series of A2[hole]^2 = c(q)^2 = 9*q^(2/3) + 18*q^(5/3) + 45*q^(8/3) + 36*q^(11/3) + 72*q^(14/3) + 54*q^(17/3) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111, Eq (63)^2.

Muniru A Asiru, Table of n, a(n) for n = 0..20000
Kevin B. Ford, Some infinite series identities, Proc. Amer. Math. Soc., Volume 119, Number 3 (November 1993), 1019-1020.
Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Cf. A033687, A096726, A097723, A134079, A144615, A242874, A281722.

Cf. A000203 (sigma), A016789 (3n+2).

sequence := List([1..100010],n->Sigma(3*n-1)/3); # Muniru A Asiru, Dec 29 2017

Basis( ModularForms( Gamma0(9), 2), 195)[3]; /* Michael Somos, Jul 14 2015 */

[SumOfDivisors(3*n+2)/3: n in [0..70]]; // Vincenzo Librandi, Jan 13 2018

with(numtheory): seq(sigma(3*n-1)/3, n=1..2000); # Muniru A Asiru, Jan 18 2018

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 26 2014 *)
Array[ DivisorSigma[1, 3 # - 1]/3 &, 69] (* Robert G. Wilson v, Jan 12 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* Michael Somos, Oct 17 2006 */

a(n)=sigma(3*n+2)/3; \\ Michel Marcus, Jul 14 2015

ModularForms( Gamma0(9), 2, prec=195).2 # Michael Somos, May 26 2014

a(n) = sigma(3*n+2)/3. Euler transform of period 3 sequence [2, 2, -4, ...]. - Vladeta Jovovic, Sep 14 2004
Expansion of q^(-2/3) * c(q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function. - Michael Somos, Oct 17 2006
Expansion of q^(-2/3) * (eta(q^3)^3 / eta(q))^2 in powers of q. - Michael Somos, Mar 16 2012
Convolution square of A033687. - Michael Somos, Oct 17 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A242874.
27 * a(n) = A096726(3*n + 2) - A281722(3*n + 2). - Michael Somos, Sep 04 2017
a(n) = A144615(n)/3. - Robert G. Wilson v, Jan 12 2018
From Peter Bala, Jan 07 2021: (Start)
a(n) = (-1)^n*A134079(n).
A(x) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(3*n+1))^2 = Sum_{n = -oo..oo} x^(4*n+2)/(1 - x^(3*n+2))^2 (apply Ford, equation 1, with c = x^(3/2), d = x^(1/2), |x| < 1 to the g.f. Sum_{n = -oo..oo} x^n /(1 - x^(3*n + 1)) of A033687).
Conjectural g.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(3*n+2))^2 = Sum_{n = -oo..oo} x^(5*n+1)/(1 - x^(3*n+1))^2. (End)
Sum_{k=1..n} a(k) = (2*Pi^2/27) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 4, -3, 6, -12, 8, -3, 13, -18, 12, -12, 14, -24, 24, -3, 18, -39, 20, -18, 32, -36, 24, -12, 31, -42, 40, -24, 30, -72, 32, -3, 48, -54, 48, -39, 38, -60, 56, -18, 42, -96, 44, -36, 78, -72, 48, -12, 57, -93, 72, -42, 54, -120, 72, -24, 80, -90, 60, -72, 62, -96, 104, -3, 84, -144, 68, -54, 96, -144, 72

Offset: 1

Michael Somos, Jun 30 2005

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

Denoted by xi(n) in Glaisher 1907. - Michael Somos, May 17 2013

			q - 3*q^2 + 4*q^3 - 3*q^4 + 6*q^5 - 12*q^6 + 8*q^7 - 3*q^8 + 13*q^9 + ...

G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(18). MR0121327 (22 #12066).
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. 8).

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).
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Cf. A000593, A008438, A046897, A096727, A112610.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ If[ Mod[ d, 4] == 0, 0, d], {d, Divisors@n}]] (* Michael Somos, May 17 2013 *)

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

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); -1/8 * polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))}

Expansion of (1 - eta(q)^8 / eta(q^2)^4) / 8 in powers of q.
a(n) = Sum_{d divides n} (-1)^(n/d + d) * d [Glaisher].
Multiplicative with a(2^e) = -3, if e>0. a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f.: Sum_{k>0} k * (x^k / (1 - x^k) - 6 * x^(2*k) / (1 - x^(2*k)) + 8 * x^(4*k) / (1 - x^(4*k))).
G.f.: Sum_{k>0} -(-x)^k / (1 + x^k)^2 = Sum_{k>0} - k * (-x)^k / (1 + x^k).
a(n) = -(-1)^n * A046897(n). a(n) = -A096727(n) / 8 unless n=0. a(2*n) = -3 * A000593(n). a(2*n + 1) = A008438(n). a(4*n + 1) = A112610(n). a(4*n + 3) = A097723(n).
Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023

A239053 Sum of divisors of 4*n-1.

Original entry on oeis.org

4, 8, 12, 24, 20, 24, 40, 32, 48, 56, 44, 48, 72, 72, 60, 104, 68, 72, 124, 80, 84, 120, 112, 120, 156, 104, 108, 152, 144, 144, 168, 128, 132, 240, 140, 168, 228, 152, 192, 216, 164, 168, 260, 248, 180, 248, 216, 192, 336, 200, 240, 312, 212, 264, 296

Offset: 1

Omar E. Pol, Mar 09 2014

Bisection of A008438.
a(n) is also the total number of cells in the n-th branch of the third quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-1), see example. For the quadrants 1, 2, 4 see A112610, A239052, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			Illustration of initial terms:
-----------------------------------------------------
.        Branches of the spiral
.        in the third quadrant             n    a(n)
-----------------------------------------------------
.     _       _       _       _
.    | |     | |     | |     | |
.    | |     | |     | |     |_|_ _
.    | |     | |     | |    2  |_ _|       1      4
.    | |     | |     |_|_     2
.    | |     | |    4    |_
.    | |     |_|_ _        |_ _ _ _
.    | |    6      |_      |_ _ _ _|       2      8
.    |_|_ _ _        |_   4
.   8      | |_ _      |
.          |_    |     |_ _ _ _ _ _
.            |_  |_    |_ _ _ _ _ _|       3     12
.           8  |_ _|  6
.                  |
.                  |_ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _|       4     24
.                 8
.
For n = 4 the sum of divisors of 4*n-1 is 1 + 3 + 5 + 15 = A000203(15) = 24. On the other hand the parts of the symmetric representation of sigma(15) are [8, 8, 8] and the sum of them is 8 + 8 + 8 = 24, equaling the sum of divisors of 15, so a(4) = 24.

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

Cf. A000203, A004767, A008438, A062731, A074400, A112610, A193553, A196020, A235791, A236104, A237270, A237591, A237593, A239050, A239052, A244050, A245092, A262626.

[SumOfDivisors(4*n-1): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016

A239053:=n->numtheory[sigma](4*n-1): seq(A239053(n), n=1..80); # Wesley Ivan Hurt, Dec 06 2016

DivisorSigma[1,4*Range[60]-1] (* Harvey P. Dale, Dec 06 2016 *)
Table[DivisorSigma[1, 4 n - 1], {n, 100}] (* Vincenzo Librandi, Dec 07 2016 *)

a(n) = sigma(4*n-1); \\ Michel Marcus, Dec 07 2016

a(n) = A000203(4n-1) = A000203(A004767(n-1)).
a(n) = 4*A097723(n-1). - Joerg Arndt, Mar 09 2014
Sum_{k=1..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

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

Offset: 0

Michael Somos, Aug 10 2006

sign

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

Number 33 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 13*x^4 - 12*x^5 + 14*x^6 - 24*x^7 + ...
G.f. = q - 4*q^3 + 6*q^5 - 8*q^7 + 13*q^9 - 12*q^11 + 14*q^13 - 24*q^15 + ...

J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.179
Hardy, et al., Collected Papers of Srinivasa Ramanujan, p. 326, Question 359.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. 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. A001938, A008438, A097723, A112610, A134343, A258831, A258835.

A := Basis( ModularForms( Gamma0(16), 2), 110); A[2] - 4*A[4]; /* Michael Somos, Jun 10 2015 */

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^2 / (4 q^(1/2)), {q, 0, n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^4, {q, 0, n}]; (* Michael Somos, Oct 14 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[1, 2 n + 1]]; (* Michael Somos, Jun 15 2015 *)

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

A = ModularForms( Gamma0(16), 2, prec=110).basis(); A[1] - 4*A[3]; # Michael Somos, Jun 27 2013

Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2))^4 in powers of q.
Expansion of q^(-1/2)/4 * k * k' * (K / (Pi/2))^2 in powers of q where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^n, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (-1)^e * (p^(e+1) - 1) / (p - 1) if p == 3 (mod 4).
Given g.f. A(x), then B(x) = 4 * Integral_{0..x} A(x^2) dx = arcsin(4 * x * A001938(x^2)) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = sin(u + v) / 2 - sin((u - v) / 2). - Michael Somos, Oct 14 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 27 2013
G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)))^4.
G.f.: Sum_{k>0} -(-1)^k * (2*k - 1) * x^(k - 1) / (1 + x^(2*k - 1)).
G.f.: (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(4*k)))^4.
G.f.: (Sum_{k>0} (-1)^floor(k/2) * x^((k^2 - k)/2))^4.
G.f.: Sum_{k>0} (-1)^k * (2*k - 1) * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A008438(n). a(2*n) = A112610(n). a(2*n + 1) = -4 * A097723(n).
Convolution square of A134343. - Michael Somos, Jun 20 2012
a(3*n + 2) = 6 * A258831(n). a(4*n + 3) = -8 * A258835(n). - Michael Somos, Jun 11 2015

A133690 Expansion of (phi(-q) * phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 8, -16, 24, -24, 32, -32, 24, -52, 48, -48, 96, -56, 64, -96, 24, -72, 104, -80, 144, -128, 96, -96, 96, -124, 112, -160, 192, -120, 192, -128, 24, -192, 144, -192, 312, -152, 160, -224, 144, -168, 256, -176, 288, -312, 192, -192, 96, -228, 248, -288

Offset: 0

Michael Somos, Sep 20 2007

sign

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

			G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 24*q^4 - 24*q^5 + 32*q^6 - 32*q^7 + 24*q^8 - ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004011, A008438, A097057, A097723, A112610, A133657, A133692.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Which[ OddQ[n], DivisorSigma[ 1, n], Mod[n, 4] > 0, -2 DivisorSigma[1, n/2], True, -6 DivisorSum[n/4, # Mod[#, 2] &]]]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n<1, n==0, -4 * if( n%2, sigma(n), n%4, -2 * sigma(n/2), -6 * sumdiv( n/4, d, (d%2)*d )))};

{a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2))^2, n))};

Expansion of (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2 in powers of q.
Euler transform of period 8 sequence [ -4, 2, -4, -8, -4, 2, -4, -4, ...].
a(n) = -4 * b(n) where b() is multiplicative with b(2) = -2, b(2^e) = -6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133657.
G.f.: ( Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2 )^2.
a(n) = (-1)^n * A097057(n). Convolution square of A133692.
a(2*n) = 8 * A046897(n) unless n=0. a(2*n + 1) = A008438(n). a(4*n) = A004011(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = -16 * A097723(n).

A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 0, -16, -8, 24, 0, -32, 24, 52, 0, -48, -32, 56, 0, -96, 24, 72, 0, -80, -48, 128, 0, -96, 96, 124, 0, -160, -64, 120, 0, -128, 24, 192, 0, -192, -104, 152, 0, -224, 144, 168, 0, -176, -96, 312, 0, -192, 96, 228, 0, -288, -112, 216, 0, -288, 192, 320, 0

Offset: 0

Michael Somos, Feb 26 2012

sign

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

			1 + 4*q - 16*q^3 - 8*q^4 + 24*q^5 - 32*q^7 + 24*q^8 + 52*q^9 - 48*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A096727, A097723, A112610, A121613, A139093, A208435.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, -q], {q, 0, n}]; Table[A207541[n], {n, 0, 50}] (* G. C. Greubel, Dec 16 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^2, n))}

Expansion of phi(-q^4)^4 + 4 * q * psi(-q^2)^4 = phi(q)^3 * phi(-q) = phi(q)^2 * phi(-q^2)^2 = psi(q)^4 * chi(-q^2)^6 = phi(-q^2)^6 / phi(-q)^2 = f(q)^6 / psi(q)^2 = f(q)^4 * chi(-q^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 4 sequence [ 4, -10, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A112610.
G.f.: Product_{k>0} (1 - x^(2*k))^14 / ((1 - x^k)^4 * (1 - x^(4*k))^6).
a(3*n + 2) = 24 * A208435(n). a(4*n + 2) = 0. a(2*n + 1) = 4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 3) = -16 * A097723(n). Convolution square of A139093.

A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, 0, -3, 4, 0, 0, 1, 0, 0, 6, -7, 0, 8, 0, -6, 8, 0, 0, -1, 0, 0, 13, -8, 0, 12, 0, -15, 12, 0, 0, 7, 0, 0, 14, -18, 0, 16, 0, -12, 24, 0, 0, -5, 0, 0, 18, -14, 0, 26, 0, -24, 20, 0, 0, 6, 0, 0, 32, -31, 0, 24, 0, -18, 24, 0, 0, 5, 0, 0, 31

Offset: 0

Michael Somos, Nov 12 2018

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 124 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = x^3 - x^4 + 2*x^6 - 3*x^8 + 4*x^9 + x^12 + 6*x^15 - 7*x^16 + ...

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..100000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A008438, A033686, A097723, A112610, A144614, A224226, A229615, A321528.

A := Basis( ModularForms( Gamma0(12), 2), 76); A[4] - A[5];

a[ n_] := SeriesCoefficient[ x^3 QPochhammer[ x, x^2] QPochhammer[ x^12]^6 / (QPochhammer[ x^3, x^6]^3 QPochhammer[ x^4]^2), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, 0, s[n/3] - s[n/4] - s[n/6] + s[n/12]]];
a[ n_] := If[ n < 1, 0, Sum[ d {0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0, 0}[[Mod[d, 12, 1]]] / 12, {d, Divisors[n]}]];

{a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0][d%12 + 1] / 12))};

{a(n) = my(s = x -> if( frac(x), 0, sigma(x))); if( n<1, 0, s(n/3) - s(n/4) - s(n/6) + s(n/12))};

{a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 * eta(x^12 + A)^6 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2), n))};

Expansion of x^3 * (psi(x^3) * psi(x^6))^3 / (psi(x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.
Expansion of x^3 * chi(-x) * f(-x^12)^6 / (chi(-x^3)^3 * f(-x^4)^2) in powers of x where chi(), f() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321528.
a(n) = s(n/3) - s(n/4) - s(n/6) + s(n/12) where s(x) = sum of divisors of x for integer x else 0.
a(6*n + 1) = a(6*n + 5) = a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A224226(n) if n>0. a(2*n) = -A229615(n). a(6*n + 3) = A008438(n). a(12*n + 6) = 2*A008438(n).
a(12*n + 3) = A112610(n). a(12*n + 4) = -A144614(n). a(12*n + 8) = -3*A033686(n). a(12*n + 9) = 4*A097723(n).

cp's OEIS Frontend

A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A097057 Number of integer solutions to a^2 + b^2 + 2*c^2 + 2*d^2 = n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

Extensions

A098098 a(n) = sigma(6*n+5)/6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A033686 One-ninth of theta series of A2[hole]^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A097057 Number of integer solutions to a^2 + b^2 + 2c^2 + 2d^2 = n.