cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A262780 Expansion of phi(-x^6) * psi(x^4) + x * phi(-x^2) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, -2, 1, 0, -2, 0, 0, 2, -2, 0, 1, 1, 0, -2, 0, 0, -2, -2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, -2, -2, 0, 2, 0, 0, 2, 1, 0, -2, 1, 0, 0, 0, 0, 4, -2, 0, 2, 0, 0, -2, 0, 0, -2, -2, 0, 0, -2, 0, 1, 0, 0, -2, 2, 0, -4, 0, 0, 2, 0, 0, 0, 3, 0, -2, 0, 0, -2, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 01 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x - 2*x^3 + x^4 - 2*x^6 + 2*x^9 - 2*x^10 + x^12 + x^13 + ...
G.f. = q + q^3 - 2*q^7 + q^9 - 2*q^13 + 2*q^19 - 2*q^21 + q^25 + ...

Links

Crossrefs

Cf. A033762, A262726, A262774.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, {1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, 1}[[Mod[n, 12, 1]]] DivisorSum[ 2 n + 1, KroneckerSymbol[ -3, #] &]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^6] EllipticTheta[ 2, 0, x^2] + EllipticTheta[ 4, 0, x^2] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (2 n + 1))];
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))};

Formula

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 19 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 13 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(2*n) = A262774(n). a(2*n + 1) = A262726(n).
abs(a(n)) = A033762(n).

A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, 2, -4, 0, 4, -8, 4, 2, 0, 0, 2, -8, 8, 0, -4, 0, 4, -8, 0, 4, 0, 0, 4, -4, 8, 2, -8, 0, 0, -8, 4, 0, 0, 0, 2, -8, 8, 4, 0, 0, 8, -8, 0, 0, 0, 0, 2, -12, 4, 0, -8, 0, 4, 0, 8, 4, 0, 0, 0, -8, 8, 4, -4, 0, 0, -8, 0, 0, 0, 0, 4, -8, 8, 2, -8, 0, 8, -8, 0, 2, 0, 0, 4, 0, 8, 0, 0, 0, 0, -16, 0, 4, 0, 0, 4, -8
Offset: 0

Views

Author

Michael Somos, Jul 29 2011

Keywords

Comments

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

Examples

			G.f. = 1 - 4*q + 4*q^2 + 2*q^3 - 4*q^4 + 4*q^6 - 8*q^7 + 4*q^8 + 2*q^9 + 2*q^12 + ...

Links

Crossrefs

Cf. A033687, A033762, A097195, A112604, A112605.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, 2 * if( n%3==1, -2, 1) * sumdiv( n, d, -(-1)^d * kronecker( -3, d)))};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)), n))};

Formula

Expansion of (-2 * a(q) +2 * a(q^2) +3 * a(q^3)) / 3 = b(q) * (b(q) + 2 * b(q^2)) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM functions.
Expansion of eta(q)^4 * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -4, -2, -2, -2, -4, -1, -4, -2, -4, -2, -4, -1, -4, -2, -2, -2, -4, -2, ...].
Moebius transform is period 18 sequence [ -4, 8, 6, -8, 4, -6, -4, 8, 0, -8, 4, 6, -4, 8, -6, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A193426.
a(3*n) = A123330(n). a(3*n + 1) = -4 * A033687(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 2) = 4 * A033687(n). a(6*n + 3) = 2 * A033762(n). a(6*n + 4) = 4 * A033687(n). a(8*n + 2) = 4 * A112604(n). a(8*n + 6) = 4 * A112605(n). a(6*n + 5) = 0. a(4*n) = a(n).

A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 6, 4, 0, 0, 6, 8, 6, 4, 0, 0, 0, 8, 12, 0, 0, 0, 6, 8, 0, 8, 0, 0, 6, 4, 12, 4, 0, 0, 0, 8, 6, 0, 0, 0, 0, 8, 12, 8, 0, 0, 12, 8, 0, 0, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 12, 8, 0, 0, 0, 8, 12, 8, 0, 0, 0, 8, 0, 0, 0, 0, 6, 8, 12, 4, 0, 0, 12, 8, 0, 4, 0, 0
Offset: 0

Views

Author

Michael Somos, Jan 06 2015

Keywords

Comments

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).

Examples

			G.f. = 1 + 4*x + 6*x^2 + 4*x^3 + 6*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + 8*x^13 + ...

Links

Crossrefs

Cf. A004016, A033687, A033762, A097195, A112604, A112605, A121361, A123884, A186706, A244339, A253625.
Cf. A000122, A000700, A005882, A005928, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(12), 1), 83); A[1] + 4*A[2] + 6*A[3] + 4*A[4];
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ (1 + Mod[k, 2]) q^k / (1 - q^k + q^(2 k)), {k, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^n Sum[(-1)^(n/d) {2, -1, 0, 1, -2, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
  • PARI
    {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, (n/d%2 + 1) * (-1)^(d\3) * (d%3>0) ))};
  • PARI
    {a(n) = if( n<1, n==0, 2 * (-1)^n * sumdiv(n, d, (-1)^(n/d) * [0, 2, -1, 0, 1, -2][d%6 + 1]))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^4 * eta(x^12 + A) / (eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3/2*(e%2), if( p==3, 1, if( p%6 == 1, e+1, 1-e%2))))))};

Formula

Expansion of phi(q)^2 * phi(-q^3)^2 / (phi(-q^2) * phi(-q^6)) = psi(q) * psi(-q^3) * (chi(q) * chi(-q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of (2*a(q) + 3*a(q^2) - 2*a(q^4)) / 3 = (b(q) - 2*b(q^4)) * (b(q) - 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^2)^8 * eta(q^3)^4 * eta(q^12) / (eta(q)^4 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 4, -4, 0, -1, 4, -4, 4, -1, 0, -4, 4, -2, ...].
Moebius transform is period 12 sequence [ 4, 2, 0, -6, -4, 0, 4, 6, 0, -2, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253625.
a(n) = 4*b(n) where b() is multiplicative with b(2^e) = (3/4) * (1 - (-1)^e) if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: 1 + Sum_{k>0} (1 + (k mod 2)) * q^k / (1 - q^k + q^(2*k)).
G.f.: Product_{k>0} (1 + q^k) * (1 - q^(2*k)) * (1 - q^(3*k)) * (1 + q^(6*k)) / ((1 + q^(2*k)) * (1 - q^k + q^(2*k)))^3.
a(n) = (-1)^n * A244339(n). a(2*n) = A004016(n). a(2*n + 1) = 4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = 4 * A097195(n). a(6*n + 2) = 6 * A033687(n). a(6*n + 4) = a(6*n = 5) = 0.
a(12*n + 1) = 4 * A123884(n). a(12*n + 2) = 6 * A097195(n). a(12*n + 3) = 4 * A112604(n). a(12*n + 7) = 8 * A121361(n). a(12*n + 9) = 4 * A112605(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Dec 30 2023

A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

0, 1, 1, -2, -1, 0, -2, 2, 1, -2, 0, 0, 2, 2, 2, 0, -1, 0, -2, 2, 0, -4, 0, 0, -2, 1, 2, -2, -2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, -4, 0, 0, -4, 2, 0, 0, 0, 0, 2, 3, 1, 0, -2, 0, -2, 0, 2, -4, 0, 0, 0, 2, 2, -4, -1, 0, 0, 2, 0, 0, 0, 0, -2, 2, 2, -2, -2, 0, -4, 2
Offset: 0

Views

Author

Michael Somos, Aug 04 2015

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Cf. A033687, A033762, A097195, A093829, A112848, A113447, A123530, A123863, A227696, A246838, A248897, A253243, A260941, A260942, A260943, A260944.
Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(36), 1), 80); A[2] + A[3] - 2*A[4] - A[5] - 2*A[7] + 2*A[8] + A[9] - 2*A[10] + 2*A[13] + 2*A[14] + 2*A[15] - A[17] - 2*A[19] - 4*A[20];
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, -1, -1, 0}[[Mod[ d, 6, 1]]] {1, 0, -2, 0, 1, 0}[[Mod[ n/d, 6, 1]]], {d, Divisors @ n}]]
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -(-1)^#2, # == 3, -2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] EllipticTheta[ 3, 0, q] / (2 q^(1/4) QPochhammer[ q^6]), {q, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 1, 0, -1, -1][d%6 + 1] * [0, 1, 0, -2, 0, 1][n\d%6 + 1]))};
  • PARI
    {a(n) = if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, -2, p%6==5, 1-e%2, e+1)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Formula

Expansion of (a(q) + a(q^2) - 3*a(q^3) - 2*a(q^4) - 3*a(q^6) + 6*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q * phi(q) * psi(-q) * psi(-q^9) / f(-q^6) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, -3, 1, -2, 1, -2, 1, -2, 0, -3, 1, -1, 1, -3, 1, -2, 1, -2, 1, -2, 1, -3, 1, -1, 1, -3, 0, -2, 1, -2, 1, -2, 1, -3, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, -3, -2, -1, 0, 1, 2, 0, 0, -1, 6, 1, 0, 3, -2, -1, 0, 1, 2, -3, 0, -1, -6, 1, 0, 0, -2, -1, 0, 1, 2, 3, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = -2, if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123863.
a(2*n) = A112848(n). a(2*n + 1) = A123530(n). a(3*n) = -2 * A113447(n). a(3*n + 1) = A227696(n).
a(4*n) = - A112848(n). a(4*n + 1) = A253243(n). a(4*n + 2) = A123530(n). a(4*n + 3) = -2 * A246838(n).
a(6*n) = -2 * A093829(n). a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n). a(6*n + 3) = -2 * A033762(n). a(6*n + 5) = 0.
a(8*n + 1) = A260941(n). a(8*n + 2) = A253243(n). a(8*n + 3) = -2 * A260943(n). a(8*n + 4) = - A123530(n). a(8*n + 5) = 2 * A260942(n). a(8*n + 6) = -2 * A246838(n). a(8*n + 7) = 2 * A260944(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A260958 Expansion of (a(q) - 3*a(q^2) + 3*a(q^3) - 4*a(q^4) + 3*a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

0, 1, -3, 4, -3, 0, 0, 2, -3, 4, 0, 0, 0, 2, -6, 0, -3, 0, 0, 2, 0, 8, 0, 0, 0, 1, -6, 4, -6, 0, 0, 2, -3, 0, 0, 0, 0, 2, -6, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, -3, 0, -6, 0, 0, 0, -6, 8, 0, 0, 0, 2, -6, 8, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, -6, 4, -6, 0, 0, 2, 0, 4
Offset: 0

Views

Author

Michael Somos, Aug 05 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = x - 3*x^2 + 4*x^3 - 3*x^4 + 2*x^7 - 3*x^8 + 4*x^9 + 2*x^13 + ...

Links

Crossrefs

Cf. A033687, A033762, A097195, A113448, A122861.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, {1, -3, 4, -3, 1, 0}[[Mod[ n, 6, 1]]] Sum[ {1, 0, 0, 0, -1, 0}[[Mod[ d, 6, 1]]], {d, Divisors @ n}]];
a[ n_] := If[ n < 1 || Mod[n, 6] == 0, 0, Times @@ (Which[ # == 1, 1, # == 2, -2 - Mod[#2, 2], # == 3, 4, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 3, 0, x^3] QPochhammer[ -x^3, x^6]^2 EllipticTheta[ 2, 0, x^(9/2)] / (2 x^(1/8) QPochhammer[-x, x^2]^2 EllipticTheta[ 3, 0, x^9]), {x, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, [0, 1, -3, 4, -3, 1][n%6+1] * sumdiv(n, d, [0, 1, 0, 0, 0, -1][n/d%6+1]))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^9 * eta(x^9 + A) * eta(x^36 + A)^2 / (eta(x^2 + A)^4 * eta(x^3 + A)^4 * eta(x^12 + A)^4 * eta(x^18 + A)^3), n))};

Formula

Expansion of q * f(q) * phi(q^3) * chi(q^3)^2 * psi(q^9) / (chi(q)^2 * phi(q^9)) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q)^3 * eta(q^4)^2 * eta(q^6)^9 * eta(q^9) * eta(q^36)^2 / (eta(q^2)^4 * eta(q^3)^4 * eta(q^12)^4 * eta(q^18)^3) in powers of q.
Euler transform of period 36 sequence [ -3, 1, 1, -1, -3, -4, -3, -1, 0, 1, -3, -2, -3, 1, 1, -1, -3, -2, -3, -1, 1, 1, -3, -2, -3, 1, 0, -1, -3, -4, -3, -1, 1, 1, -3, -2, ...].
Moebius transform is period 36 sequence [ 1, -4, 3, 0, -1, 0, 1, 0, 0, 4, -1, 0, 1, -4, -3, 0, -1, 0, 1, 0, 3, 4, -1, 0, 1, -4, 0, 0, -1, 0, 1, 0, -3, 4, -1, 0, ...].
a(2*n) = A113448(n). a(3*n + 1) = A122861(n). a(6*n) = 0. a(6*n + 1) = A097195(n). a(6*n + 2) = a(12*n + 4) = -3 * A033687(n). a(6*n + 3) = 4 * A033762(n). a(6*n + 5) = a(12*n + 10) = 0.

A261884 Expansion of (a(q) - a(q^2) - 2*a(q^3) + 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

Original entry on oeis.org

1, -1, -1, 1, 0, 1, 2, -1, -1, 0, 0, -1, 2, -2, 0, 1, 0, 1, 2, 0, -2, 0, 0, 1, 1, -2, -1, 2, 0, 0, 2, -1, 0, 0, 0, -1, 2, -2, -2, 0, 0, 2, 2, 0, 0, 0, 0, -1, 3, -1, 0, 2, 0, 1, 0, -2, -2, 0, 0, 0, 2, -2, -2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 2, -2, -1, 2, 0, 2, 2, 0
Offset: 1

Views

Author

Michael Somos, Sep 04 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A033687, A033762, A093829, A097195, A035178 (apparently gives the absolute values).
Cf. A004016, A005882, A005928.

Programs

  • Mathematica
    A004016[q_] := (QPochhammer[q]^3 + 9*q*QPochhammer[q^9]^3)/ QPochhammer[q^3]; A261884[n_] := SeriesCoefficient[(A004016[q] - A004016[q^2] - 2*A004016[q^3] + 2*A004016[q^6])/6, {q, 0, n}]; Table[A261884[n], {n, 1, 50}] (* G. C. Greubel, Sep 24 2017 *)
  • PARI
    {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, (-1)^e, p==3, -1, p%6==1, e+1, 1-e%2)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) - x * eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};

Formula

Moebius transform is period 18 sequence [ 1, -2, -2, 2, -1, 4, 1, -2, 0, 2, -1, -4, 1, -2, 2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} F(x^(6*k - 5)) - F(x^(6*k - 3)) + F(x^(6*k - 1)) where F(x) := x / (1 + x + x^2).
a(n) = A093829(n) unless n == 0 (mod 3). a(2*n) = - a(n). a(3*n + 1) = A033687(n).
a(6*n + 1) = A097195(n). a(6*n + 3) = - A033762(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(18*sqrt(3)) = 0.100766631346... . - Amiram Eldar, Nov 23 2023

A286813 Number of positive odd solutions to equation x^2 + 8*y^2 = 8*n + 9.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 2, 0
Offset: 0

Views

Author

Seiichi Manyama, May 28 2017

Keywords

Links

Crossrefs

Related to the number of positive odd solutions to equation x^2 + k*y^2 = 8*n + k + 1: A008441 (k=1), A033761 (k=2), A033762 (k=3), A053692 (k=4), A033764 (k=5), A259896 (k=6), A035162 (k=7), this sequence (k=8).

Formula

Expansion of q^(-9/8) * (eta(q^2) * eta(q^16))^2 / (eta(q) * eta(q^8)) in powers of q.
Euler Transform of -(-2*x^8-x^7-1)/(x^9+x^8+x+1) (o.g.f.). - Simon Plouffe, Jun 23 2018

A329958 Expansion of q^(-13/24) * eta(q^2)^3 * eta(q^3) * eta(q^6) / eta(q)^2 in powers of q.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 3, 5, 3, 6, 7, 4, 5, 4, 8, 6, 5, 7, 6, 7, 8, 7, 5, 8, 10, 9, 4, 7, 7, 9, 11, 8, 10, 5, 10, 12, 7, 10, 8, 10, 12, 4, 10, 8, 13, 15, 10, 9, 5, 15, 9, 12, 11, 10, 12, 10, 11, 11, 12, 15, 12, 6, 14, 8, 11, 17, 13, 12, 9, 16, 17, 8, 15, 10, 14
Offset: 0

Views

Author

Michael Somos, Nov 26 2019

Keywords

Examples

			G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + ...
G.f. = q^13 + 2*q^37 + 2*q^61 + 3*q^85 + 3*q^109 + 4*q^133 + 4*q^157 + ...

Crossrefs

Cf. A010054, A033762, A080995, A121444, A329955.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 QPochhammer[ x^3] QPochhammer[ x^6] / QPochhammer[ x]^2, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^2, n))};

Formula

Euler transform of period 6 sequence [2, -1, 1, -1, 2, -3, ...].
G.f.: Product_{k>=1} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
Convolution of A033762 and A080995. Convolution of A010054 and A121444.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (3/2)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329955.
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