A097723 -id:A097723 - cp's OEIS frontend

A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, 4, 4, 0, 6, 16, 8, 0, 13, 24, 12, 0, 14, 32, 24, 0, 18, 52, 20, 0, 32, 48, 24, 0, 31, 56, 40, 0, 30, 96, 32, 0, 48, 72, 48, 0, 38, 80, 56, 0, 42, 128, 44, 0, 78, 96, 48, 0, 57, 124, 72, 0, 54, 160, 72, 0, 80, 120, 60, 0, 62, 128, 104, 0, 84, 192, 68, 0, 96

Offset: 1

Michael Somos, Sep 20 2007

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

			G.f. = q + 4*q^2 + 4*q^3 + 6*q^5 + 16*q^6 + 8*q^7 + 13*q^9 + 24*q^10 + ...

Amiram Eldar, 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. A008438, A097723, A112610, A121455, A133690, A222068.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]/2)^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[n_] := Switch[IntegerExponent[n, 2], 0, DivisorSigma[1, n], 1, 4*DivisorSigma[1, n/2], , 0]; Array[a, 100] (* _Amiram Eldar, Nov 12 2022 *)

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

{a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3) )^2, n))};

Expansion of (eta(q^2)^5 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 8 sequence [ 4, -6, 4, 0, 4, -6, 4, -4, ...].
a(n) is multiplicative with a(2) = 4, a(2^e) = 0 if e>1, a(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)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133690.
a(4*n) = 0. a(4*n+2) = 4 * sigma(2*n+1). a(2*n+1) = sigma(2*n+1).
a(n) = -(-1)^n * A121455(n). Convolution square of A113411.
a(2*n + 1) = A008438. a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/16 = 0.6168502... (A222068). - Amiram Eldar, Nov 12 2022

A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -6, 6, -8, 8, -6, 13, -12, 12, -24, 14, -16, 24, -6, 18, -26, 20, -36, 32, -24, 24, -24, 31, -28, 40, -48, 30, -48, 32, -6, 48, -36, 48, -78, 38, -40, 56, -36, 42, -64, 44, -72, 78, -48, 48, -24, 57, -62, 72, -84, 54, -80, 72, -48, 80, -60, 60, -144

Offset: 1

Michael Somos, Sep 20 2007

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 - 6*q^4 + 6*q^5 - 8*q^6 + 8*q^7 - 6*q^8 + 13*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. A000593, A008438, A046897, A097723, A111973, A112610, A133690.

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

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

Expansion of (1 - (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2) / 4 in powers of q.
a(n) is multiplicative with a(2) = -2, a(2^e) = -6 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
a(n) = -4 * A133690(n) = -(-1)^n * A111973(n). a(2*n) = -2 * A046897(n). a(2*n + 1) = A008438(n). a(4*n) = -6 * A000593(n). a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).
Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 5/2^s + 1/2^(2*s-1) + 1/2^(3*s-3)). - Amiram Eldar, Oct 28 2023

A208451 Expansion of phi(q) * phi(-q)^3 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

Offset: 0

Michael Somos, Feb 26 2012

sign

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

			G.f. = 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. A004011, A008438, A096727, A097723, A112610, A121613.

A := Basis( ModularForms( Gamma0(16), 2), 58); A[1] - 4*A[2] + 16*A[4] - 8*A[5]; /* Michael Somos, Aug 21 2014 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4])^2, {q, 0, n}]; (* Michael Somos, Aug 21 2014 *)

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

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

Expansion of phi(-q^4)^4 - 4 * q * psi(-q^2)^4 = phi(q) * phi(-q)^3 = phi(-q)^2 * phi(-q^2)^2 = phi(-q^2)^6 / phi(q)^2 = psi(-q)^4 * chi(-q^2)^6 = f(-q)^4 * chi(-q^2)^2 = f(-q)^6 / psi(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (eta(q)^2 * eta(x^2) / eta(x^4))^2 in powers of q.
Euler transform of period 4 sequence [ -4, -6, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 5128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A097723.
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). a(8*n) = A004011(n). a(8*n + 4) = -8 * A008438(n).

A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 2, 4, 4, 5, 6, 7, 10, 9, 12, 11, 14, 16, 14, 15, 16, 20, 22, 19, 20, 21, 22, 31, 28, 28, 26, 30, 34, 29, 30, 36, 32, 40, 38, 35, 36, 37, 56, 39, 40, 41, 42, 52, 48, 57, 50, 47, 62, 49, 50, 56, 60, 64, 54, 55, 62, 57, 70, 68, 60, 66, 62, 76, 70, 70, 76

Offset: 1

M. F. Hasler, Dec 08 2017

Robert G. Wilson v observes in A280098 that {1, 3, 4, 6, 8, 12, 24} seem to be the only positive integers k such that sigma(kn-1)/k is an integer for all n > 0.

Muniru A Asiru, Table of n, a(n) for n = 1..100000

Cf. A280098 (analog for k = 24), A097723 (analog for k = 4), A033686 (analog for k = 3), A000203 (sigma, also the  analog for k = 1).
The analog for k = 8 is A258835, up to the offset.
The analog for k = 6 is A098098 (up to the offset), a signed variant of this and the preceding one is A258831.
Cf. A086463.

sequence := List([1..10^5], n-> Sigma(12 *n-1)/12); # Muniru A Asiru, Dec 28 2017

with(numtheory):
seq(sigma(12*n-1)/12, n=1..10^3); # Muniru A Asiru, Dec 28 2017

Array[DivisorSigma[1, 12 # - 1]/12 &, 66] (* Michael De Vlieger, Dec 08 2017 *)

PARI
```
vector(90,n,sigma(12*n-1)/12)
```

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

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A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

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A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

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A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

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A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

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