A134077 -id:A134077 - cp's OEIS frontend

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

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)

A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, -3, 15, -3, -18, 15, -24, -3, 69, -18, -36, 15, -42, -24, 90, -3, -54, 69, -60, -18, 120, -36, -72, 15, -93, -42, 231, -24, -90, 90, -96, -3, 180, -54, -144, 69, -114, -60, 210, -18, -126, 120, -132, -36, 414, -72, -144, 15, -171, -93, 270, -42, -162

Offset: 0

Michael Somos, Jul 30 2007

sign

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

			G.f. = 1 - 3*q - 3*q^2 + 15*q^3 - 3*q^4 - 18*q^5 + 15*q^6 - 24*q^7 - 3*q^8 +...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).

Cf. A121443, A131944, A134077, A168611.

A := Basis( ModularForms( Gamma0(6), 2), 54); A[1] - 3*A[2] - 3*A[3]; /* Michael Somos, Aug 30 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^2]^3 / (QPochhammer[ q^3] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Nov 21 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -3 Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<1, n==0, -3 * sumdiv(n, d, d * ((d%6==1) + (d%6==5) - 2 * (d%6==3))))};

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

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1) )))}; /* Michael Somos, Nov 21 2013 */

A = ModularForms( Gamma0(6), 2, prec=54) . basis();  A[0] - 3*A[1] - 3*A[2]; # Michael Somos, Nov 21 2013

Expansion of eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -3, -6, -2, -6, -3, -4, ...].
a(n) = -3 * b(n) where b() is multiplicative with b(2^e) = 1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Nov 21 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 54 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121443.
G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k))).
G.f.: 1 - 3 * (Sum_{k>0} (6*k - 1) * x^(6*k - 1) / (1 - x^(6*k - 1)) - 2*(6*k - 5) * x^(6*k - 3) / (1 - x^(6*k - 3)) + (6*k - 5) * x^(6*k - 5) / (1 -x^(6*k - 5))).
a(n) = a(2*n). a(n) = -3 * A131944(n) unless n=0. a(3^n) = 3 * A168611(n+1). a(2*n + 1) = -3 * A134077(n). - Michael Somos, Nov 21 2013

A133739 Expansion of q * (psi(q^6) / psi(q^3))^3 * phi(q)^5 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 9, 31, 45, 6, -45, 8, 117, 121, 54, 12, -9, 14, 72, 186, 261, 18, -207, 20, 270, 248, 108, 24, 63, 31, 126, 391, 360, 30, -270, 32, 549, 372, 162, 48, -171, 38, 180, 434, 702, 42, -360, 44, 540, 726, 216, 48, 207, 57, 279, 558, 630, 54, -693, 72, 936, 620

Offset: 1

Michael Somos, Oct 06 2007

sign

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

			G.f. = q + 9*q^2 + 31*q^3 + 45*q^4 + 6*q^5 - 45*q^6 + 8*q^7 + 117*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A098098, A134077, A134078.

A := Basis( ModularForms( Gamma0(12), 2), 58); A[2] + 9*A[3] + 31*A[4] + 45*A[5]; /* Michael Somos, Oct 30 2015 */

a[ n_] := SeriesCoefficient[ 2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(3/2)])^3 (EllipticTheta[ 3, 0, x]^5 / EllipticTheta[ 2, 0, x^(1/2)]), {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
QP=QPochhammer; CoefficientList[Series[QP[q^2]^23*QP[q^3]^3*QP[q^12]^6/( QP[q]^9*QP[q^4]^10*QP[q^6]^9), {q,0,50}],q] (* G. C. Greubel, Nov 16 2018 *)

{a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^23 * eta(x^3 + A)^3 * eta(x^12 + A)^6 / (eta(x + A)^9 * eta(x^4 + A)^10 * eta(x^6 + A)^9), n))};

Expansion of eta(q^2)^23 * eta(q^3)^3 * eta(q^12)^6 / (eta(q)^9 * eta(q^4)^10 * eta(q^6)^9) in powers of q.
Euler transform of period 12 sequence [ 9, -14, 6, -4, 9, -8, 9, -4, 6, -14, 9, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134078.
G.f.: f(x) + 6 * f(x^2) + 27 * f(x^3) + 20 * f(x^4) - 162 * f(x^6) + 108 * f(x^12) where f() is the g.f. of A000203.
a(4*n + 2) = 9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, -3, 5, -3, -6, 15, -8, -3, 23, -18, -12, 15, -14, -24, 30, -3, -18, 69, -20, -18, 40, -36, -24, 15, -31, -42, 77, -24, -30, 90, -32, -3, 60, -54, -48, 69, -38, -60, 70, -18, -42, 120, -44, -36, 138, -72, -48, 15, -57, -93, 90, -42, -54, 231, -72, -24, 100

Offset: 0

Michael Somos, Jul 18 2012

sign

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

			1 - q - 3*q^2 + 5*q^3 - 3*q^4 - 6*q^5 + 15*q^6 - 8*q^7 - 3*q^8 + 23*q^9 + ...

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. A118271, A131943, A134077, A214361.

a[ n_] := SeriesCoefficient[ (9 EllipticTheta[ 3, 0, q^3]^4 - EllipticTheta[ 3, 0, q]^4) / 8, {q, 0, n}]
a[ n_] := SeriesCoefficient[ QPochhammer[q^2]^4 QPochhammer[-q^3, q^6]^3 /QPochhammer[-q, q^2], {q, 0, n}]
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(1/2)]^4 / (16 (-q)^(1/2)), {q, 0, n}]

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

{a(n) = if( n<1, n==0, -sigma(n) + if( n%3==0, 9 * sigma(n/3)) + if( n%4==0, 4 * sigma(n/4)) + if( n%12==0, -36 * sigma(n/12)))}

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1))))))}

Expansion of (9 * phi(q^3)^4 - phi(q)^4) / 8 = phi(q) * (psi(-q) * chi(q^3))^3 = psi(-q)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^2)^2 * eta(q^4) * eta(q^6)^6 / ( eta(q^3) * eta(q^12))^3 in powers of q.
Euler transform of period 12 sequence [ -1, -3, 2, -4, -1, -6, -1, -4, 2, -3, -1, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 36 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A214361.
a(n) = -b(n) where b() is multiplicative with b(2^e) = 3 if e>0, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)).
a(n) = (-1)^n * A118271(n). a(2*n) = a(4*n) = A131943(n). a(2*n + 1) = -A134077(n).

A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, -3, -15, -3, 18, 15, 24, -3, -69, -18, 36, 15, 42, -24, -90, -3, 54, 69, 60, -18, -120, -36, 72, 15, 93, -42, -231, -24, 90, 90, 96, -3, -180, -54, 144, 69, 114, -60, -210, -18, 126, 120, 132, -36, -414, -72, 144, 15, 171, -93, -270, -42, 162, 231, 216

Offset: 0

Michael Somos, May 27 2013

sign

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

Number 21 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			1 + 3*q - 3*q^2 - 15*q^3 - 3*q^4 + 18*q^5 + 15*q^6 + 24*q^7 - 3*q^8 + ...

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
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A131943, A134077, A226132.

a[ n_] := If[ n < 1, Boole[n == 0], - 3 (-1)^n Sum[ d {0, 1, 0, -2, 0, 1}[[Mod[ d, 6] + 1]], {d, Divisors @n}]]
a[ n_] := If[ n < 2, Boole[n == 0] + 3 Boole[n == 1], 3 Times @@ (Which[ # == 2, -1, # == 3, 4 - 3 #^#2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)] / 4, {q, 0, n}]

{a(n) = if( n<1, n==0, -3 * (-1)^n * sumdiv( n, d, d * [0, 1, 0, -2, 0, 1][d%6 + 1]))}

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 3 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -1, if( p==3, 4 - p^(e+1), (p^(e+1) - 1) / (p - 1))))))}

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

Expansion of = b(q^2) * (2 * b(q^4) - b(q)) = b(q^2)^4 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM function.
Expansion of (a(q) + a(q^2)) * (a(q^2) - 2 * a(q^4)) / 2 in powers of q where a() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3)) * (phi(-q^2)^3 / phi(-q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
a(n) = 3 * b(n) where b(n) is multiplicative and b(2^e) = -1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226132.
Euler transform of period 12 sequence [ 3, -9, 2, -6, 3, -6, 3, -6, 2, -9, 3, -4, ...].
G.f.: 1 + 3 * (Sum_{k>0} x^k / (1 - (-x)^k)^2 - 9 * x^(3*k) / (1 - (-x)^(3*k))^2).
G.f.: 1 + 3 * Sum_{k>0 not 2|k} k * (x^k * x^k / (1 + x^(2*k)) - 9 * x^(3*k) / (1 + x^(3*k))).
G.f.: Product_{k>0} (1 - x^(2*k))^12 * (1 + x^(6*k)) / ( (1 - x^k)^3 * (1 + x^(3*k))^3 * (1 - x^(3*k))^2 * (1 - x^(4*k))^3 ).
a(n) = (-1)^n * A131943(n). a(2*n) = A131943(n). a(2*n + 1) = 3 * A134077(n).

A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

Original entry on oeis.org

1, -6, 6, 30, -66, -36, 186, -48, -210, 138, 36, -72, 114, -84, 48, 180, -498, -108, 726, -120, -396, 240, 72, -144, -30, -186, 84, 462, -528, -180, 1116, -192, -1074, 360, 108, -288, 654, -228, 120, 420, -1260, -252, 1488, -264, -792, 828, 144, -288, -318

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 64 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = 1 - 6*x + 6*x^2 + 30*x^3 - 66*x^4 - 36*x^5 + 186*x^6 - 48*x^7 + ...

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. A098098, A134077, A321527.

A := Basis( ModularForms( Gamma0(12), 2), 49); A[1] - 6*A[2] + 6*A[3] + 30*A[4] - 66*A[5];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^2])^3 / ( EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n==0], -6 (s[n/1] - 4 s[n/2] - 9 s[n/3] + 16 s[n/4])]];
a[ n_] := If[ n < 1, Boole[n==0], -6 Sum[ d {1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0}[[Mod[d, 12, 1]]], {d, Divisors[n]}]];

{a(n) = if( n<1, n==0, -6 * sumdiv( n, d, d * [0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1][d%12 + 1]))};

{a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)))};

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

Expansion of phi(-x) * phi(-x^2)^3 / (phi(-x^3) * phi(-x^6)) in powers of x where phi() is a Ramanujan theta function.
Expansion of eta(q)^6 * eta(q^2)^3 * eta(q^12) / (eta(q^3)^2 * eta(q^4)^3 * eta(q^6)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 864 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321527.
a(n) = -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)) if n>0, where s(x) = sum of divisors of x for integer x else 0.
a(2*n + 1) = -6 * A134077(n). a(6*n + 5) = -a(12*n + 10) = -36 * A098098(n).

A329651 Expansion of x * (psi(x^6) / psi(-x^3))^3 * phi(-x)^5 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

0, 1, -9, 31, -45, 6, 45, 8, -117, 121, -54, 12, 9, 14, -72, 186, -261, 18, 207, 20, -270, 248, -108, 24, -63, 31, -126, 391, -360, 30, 270, 32, -549, 372, -162, 48, 171, 38, -180, 434, -702, 42, 360, 44, -540, 726, -216, 48, -207, 57, -279, 558, -630, 54, 693

Offset: 0

Michael Somos, Nov 18 2019

sign

Number 105 of the 126 eta-quotients listed in Table 1 of Williams 2012.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A328788.

			G.f. = x - 9*x^2 + 31*x^3 - 45*x^4 + 6*x^5 + 45*x^6 + 8*x^7 - 117*x^8 + ...

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. A000203, A098098, A133739, A134077, A328788.

A := Basis( ModularForms( Gamma0(12), 2), 52); A[2] - 9*A[3] + 31*A[4] - 45*A[5];

a[ n_] := SeriesCoefficient[ 1/2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)])^3 EllipticTheta[ 4, 0, x]^5 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}] // PowerExpand;

{a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, 0, s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4))};

{a(n) = my(A); if( n < 1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^9 * eta(x^12 + A)^3 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^4 + A)), n))};

Euler transform of period 12 sequence [-9, -5, -6, -4, -9, -2, -9, -4, -6, -5, -9, -4, ...].
Expansion of x * phi(-x)^5 / psi(-x) * (f(-x^12) / f(-x^3))^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q)^9 * eta(q^12)^3 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^4)) in powers of q.
a(n) = s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(n) = -(-1)^n * A133739(n). a(4*n + 2) = -9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

cp's OEIS Frontend

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

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A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

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A133739 Expansion of q * (psi(q^6) / psi(q^3))^3 * phi(q)^5 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

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A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

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A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

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A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

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