A053694 -id:A053694 - cp's OEIS frontend

A094247 Expansion of (phi(-q^5)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

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

Offset: 1

Michael Somos, Apr 24 2004

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

			G.f. = q - q^2 - q^4 + q^5 - q^8 + q^9 - q^10 + 2*q^13 - q^16 + 2*q^17 - q^18 - q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 313 eq. (2.4).
L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345; see p. 336 eq. (3.13), p. 338 eq. (3.21).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053694, A122190, A214316.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5] QPochhammer[ q^20] QPochhammer[q, q^2], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^5 + A)),n))};

{a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, kronecker( -100, d)))}; /* Michael Somos, Aug 24 2006 */

Expansion of q * f(q^5) * f(-q^20) * chi(-q) in powers of q where f() and chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^5)) in powers of q.
Euler transform of period 10 sequence [-1, 0, -1, 0, 0, 0, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e > 0. a(5^e) = 1, a(p^e) = e+1 if p == 1, 5 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A214316. - Michael Somos, Jul 12 2012
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(10*k))^3 / ((1 - x^(2*k)) * (1 - x^(5*k))).
G.f.: Sum_{k>0} Kronecker( -100, k) * x^k / (1 + x^k) = Sum_{k>0} Kronecker( -25, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)). - Michael Somos, Jul 12 2012
a(n+1) = (-1)^n * A053694(n). a(4*n + 1) = A122190(n).
a(4*n + 3) = 0. a(2*n) = - A053694(n). - Michael Somos, Jul 12 2012

A122190 Expansion of q^(-1/4) * eta(q^2) * eta(q^5)^3 / (eta(q) * eta(q^10)) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 24 2006

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

Convolution product of A133100 and A133101. - Michael Somos, Feb 10 2015

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + x^11 + ...
G.f. = q + q^5 + q^9 + 2*q^13 + 2*q^17 + q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A019694, A053694, A094247, A133100, A133101, A133573.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^5]^3 / (QPochhammer[ x] QPochhammer[ x^10]), {x, 0, n}]; (* Michael Somos, Feb 10 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A)^3 / (eta(x + A) * eta(x^10 + A)), n))};

{a(n) = my(A, p, e); if( n<0, 0, n = 4*n + 1; A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==5, 1, if( p%4==1, e+1, (1 + (-1)^e) / 2))))))};

Euler transform of period 10 sequence [ 1, 0, 1, 0, -2, 0, 1, 0, 1, -2, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(5^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f.: Product_{k>0} (1 - x^(5*k))^2 * (1 + x^k) / (1 + x^(5*k)).
G.f.: Sum_{k>=0} a(k) * x^(4k+1) = Sum_{k>0 odd} x^k * (1 - x^(2*k)) * (1 - x^(6*k)) / (1 + x^(10*k)).
Expansion of f(x, x^4) * f(x^2, x^3) in powers of x where f() is the Ramanujan two-variable theta function.
Expansion of psi(x)^2 - x * psi(x^5)^2 in powers of x where psi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133573.
a(n) = A053694(4*n) = A094247(4*n + 1).
a(3*n + 2) = a(5*n + 1) = a(n). - Michael Somos, Feb 10 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/5 = 1.256637... (A019694). - Amiram Eldar, Dec 29 2023

A133574 Expansion of (5 * phi(q^5)^2 - phi(q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -1, 0, -1, 3, 0, 0, -1, -1, 3, 0, 0, -2, 0, 0, -1, -2, -1, 0, 3, 0, 0, 0, 0, 7, -2, 0, 0, -2, 0, 0, -1, 0, -2, 0, -1, -2, 0, 0, 3, -2, 0, 0, 0, 3, 0, 0, 0, -1, 7, 0, -2, -2, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 6, 0, 0, -2, 0, 0, 0, -1, -2, -2, 0, 0, 0

Offset: 0

Michael Somos, Sep 17 2007

sign

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

			G.f. = 1 - q - q^2 - q^4 + 3*q^5 - q^8 - q^9 + 3*q^10 - 2*q^13 - q^16 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A053694, A133573.

Cf. A000700, A000122, A010054, A121373.

A := Basis( ModularForms( Gamma1(20), 1), 78); A[1] - A[2] - A[3] - A[5] + 3*A[6] - A[9]; /* Michael Somos, Oct 31 2015 */

a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 3, 0, q^5]^2 - EllipticTheta[ 3, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ -q^5, q^10] / QPochhammer[ -q, q^2], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/4) EllipticTheta[ 2, Pi/4, x^(1/2)]^2 QPochhammer[ -x, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := If[n < 1, Boole[n == 0], DivisorSum[ n, If[Mod[#, 5] == 0, 5 KroneckerSymbol[-4, #/5], 0] - KroneckerSymbol[-4, #] &]]; (* Michael Somos, Oct 31 2015 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, if( d%5==0, kronecker( -4, d/5) * 5) - kronecker( -4, d)))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod(k = 1, matsize(A) [1], [p, e] = A[k, ]; if(p == 2, 1, p == 5, 1 - 4*e, p%4 == 1, e+1, 1-e%2 )))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^5 + A) * eta(x^20 + A)), n))};

Expansion of psi(-q)^2 * chi(q) * chi(q^5) in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^10)^2 / (eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, -1, -2, -1, -1, 0, -2, -1, -1, -1, -2, ...].
Moebius transform is period 20 sequence [ -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4), A(x^8)) where f(u1, u2, u4, u8) = (u1 - u2)^2 * (u4 - 2*u8)^2 - u2 * u4 * (u2 - u4) * (u2 - 2*u4).
a(n) = -b(n) where b() is multiplicative with b(2^e) = 1, b(5^e) = 1-4*e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 10 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053694.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).
G.f.: 1 - (Sum_{k>0} x^k / (1 + x^(2*k)) - 5 * x^(5*k) / (1 + x^(10*k))).
a(n) = (-1)^n * A133573(n).
Sum_{k=1..n} abs(a(k)) ~ (8*Pi/25) * n. - Amiram Eldar, Jan 27 2024

A185276 Kronecker symbol (-100 / n).

Original entry on oeis.org

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

Offset: 0

Michael Somos, Feb 19 2011

This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and A289741 = {(-20/n)}. - Jianing Song, Nov 14 2024

			x - x^3 - x^7 + x^9 - x^11 + x^13 + x^17 - x^19 + x^21 - x^23 - x^27 + ...

Jianing Song, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients, 2014.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,-1).

Cf. A000012, A053694, A289741.

f[n_] := KroneckerSymbol[-100, n]; Array[f, 100] (* Robert G. Wilson v *)

PARI
```
{a(n) = kronecker( -100, n)}
```

{a(n) = (n%2) * (-1) ^ (n\10) * kronecker( 5, n)}

{a(n) = sign(n) * polcoeff( x * (1 - x^2) * (1 - x^6) / (1 + x^10) + x * O(x^abs(n)), abs(n))}

{a(n) = local( A, p, e); if( n==0, 0, A = factor( abs(n)); sign(n) * prod( k=1, matsize( A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2 || p==5, 0, if( p%4==1, 1, (-1)^e )))))}

a(n) is multiplicative with a(2^e) = a(5^e) = 0^e, a(p^e) = 1 if p == 1 (mod 4) and p>5, a(p^e) = (-1)^e if p == 3 (mod 4).
Euler transform of length 20 sequence [ 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1].
G.f.: x * (1 - x^2) * (1 - x^6) / (1 + x^10) = x / (1 + x^2) - x^5 / (1 + x^10).
a(n + 20) = -a(-n) = a(n). a(2*n) = a(5*n) = 0.
Dirichlet convolution with A000012 is A053694 offset 1.
Sum_{k=1..n} abs(a(k)) ~ 2*n/5. - Amiram Eldar, Jan 29 2024

a(0) prepended by Jianing Song, Nov 14 2024

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A094247 Expansion of (phi(-q^5)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

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A122190 Expansion of q^(-1/4) * eta(q^2) * eta(q^5)^3 / (eta(q) * eta(q^10)) in powers of q.

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A133574 Expansion of (5 * phi(q^5)^2 - phi(q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

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A185276 Kronecker symbol (-100 / n).

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