A257398 -id:A257398 - cp's OEIS frontend

A257399 Expansion of phi(x^3) * phi(-x^12) / chi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.

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

Offset: 0

Michael Somos, Apr 21 2015

nonn

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

			G.f. = 1 + 2*x^3 + x^4 + 2*x^7 + x^8 + 2*x^11 + 2*x^12 + 2*x^16 + 3*x^20 + ...
G.f. = q + 2*q^19 + q^25 + 2*q^43 + q^49 + 2*q^67 + 2*q^73 + 2*q^97 + ...

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. A204531, A255317, A257398, A257400.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^12], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^8 + A) / (eta(x^4 + A) * eta(x^3 + A)^2 * eta(x^24 + A)), n))};

Expansion of (phi(-x^24)^2 + 2 * x^3 * psi(-x^12)^2) / chi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/6) * eta(q^6)^5 * eta(q^8) / (eta(q^4) * eta(q^3)^2 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 0, 0, 2, 1, 0, -3, 0, 0, 2, 0, 0, -2, 0, 0, 2, 0, 0, -3, 0, 1, 2, 0, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257400.
a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A257398(n). a(4*n + 3) = 2 * A255317(n).

A257403 Multiplicative with a(2) = 1, a(2^e) = 0 if e>1, a(3^e) = 0^e, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 21 2015

			G.f. = x + x^2 + 2*x^11 + 2*x^17 + 2*x^19 + 2*x^22 + x^25 + 2*x^34 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A255317, A255318, A255319, A256505, A256574, A257398, A257399, A257402, A257477.

a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ # == 2, Boole[#2 == 1], # == 3, 0, Mod[#, 8] < 4, #2 + 1, True, Mod[#2 + 1, 2]]& @@@ FactorInteger[n])];

{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<5, p+e==3, p%8 > 4, 1-e%2, e+1)))};

Moebius transform is the period 288 sequence A257477.
a(3*n) = a(4*n) = a(8*n + 5) = a(8*n + 7) = 0. a(2*n + 1) = a(4*n + 2).
a(6*n + 1) = A257399(n). a(6*n + 5) = 2*A257402(n).
a(24*n + 1) = A257398(n). a(24*n + 11) = 2*A255318(n). a(24*n + 17) = 2*A255319(n). a(24*n + 19) = 2*A255317(n).
From Michael Somos, Apr 22 2015: (Start)
a(3*n + 2) = A256505(n) unless n == 5 (mod 8). a(3*n + 19) = 2 * A256574(n) unless n == 2 (mod 8).
Expansion of F(q) + F(q^2) + G(q) + G(q^2) in powers of q where F(q) = q * A257399(q^6) and G(q) = 2 * q^11 * A257402(q^6). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(2)) = 0.370240... . - Amiram Eldar, Oct 17 2022

A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 - x - x^5 - 2*x^6 + 2*x^7 + x^8 + 2*x^11 - 2*x^14 + x^16 - x^21 + ...
G.f. = q - q^4 - q^16 - 2*q^19 + 2*q^22 + q^25 + 2*q^34 - 2*q^43 + q^49 + ...

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

Cf. A082564, A129134, A255318, A255319, A257398.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ 3 n, 2] DivisorSum[ 3 n + 1, KroneckerSymbol[-2, #] &]];

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

{a(n) = if( n<0, 0, (-1)^(3*n\2) * sumdiv(3*n + 1, d, kronecker( -2, d)))};

Expansion of q^(-1/3) * eta(q) * eta(q^6)^4 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, 0, -1, -3, -1, 0, 0, 0, -1, -2, ...].
G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k))^2 / ( (1 + x^k) * (1 + x^(6*k)) ).
-2 * a(n) = A082564(3*n + 1). a(n) = A129134(3*n + 1).
a(4*n + 3) = 2 * A257402(n-1). a(8*n) = A257398(n). a(8*n + 2) = a(8*n + 4) = a(16*n + 3) = a(16*n + 15) = 0. a(16*n + 7) = 2 * A255318(n). a(16*n + 11) = 2 * A255319(n).

A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 - x^2 - 2*x^3 + 2*x^5 - x^10 + 2*x^12 + 2*x^13 - 2*x^14 + x^16 + ...
G.f. = q^2 - q^8 - 2*q^11 + 2*q^17 - q^32 + 2*q^38 + 2*q^41 - 2*q^44 + ...

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

Cf. A082564, A129134, A255317, A255318, A255319, A257399, A257402.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] QPochhammer[ x^3]^2 / QPochhammer[ x^6, x^12]^2, {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ n, 2] DivisorSum[ 3 n + 2, KroneckerSymbol[-2, #] &]];

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

{a(n) = if( n<0, 0, (-1)^(n\2) * sumdiv(3*n + 2, d, kronecker( -2, d)))};

Expansion of q^(-2/3) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 0, -1, -2, 0, 0, -1, 0, 0, -2, -1, 0, -2, ...].
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2.
a(n) = A129134(3*n + 2). -2 * a(n) = A082564(3*n + 2).
a(4*n) = A257399(n). a(8*n + 3) = -2 * A255318(n). a(8*n + 5) = 2 * A255319(n). a(8*n + 6) = -2 * A257402(n-1). a(16*n) = A257398(n). a(16*n + 2) = - A257399(n). a(16*n + 12) = 2 * A255317(n).
a(8*n + 1) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.

A256505 Expansion of phi(x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 22 2015

nonn

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

			G.f. = 1 + 2*x^3 + 2*x^12 + x^16 + 2*x^19 + 2*x^27 + 2*x^28 + x^32 + ...
G.f. = q^2 + 2*q^11 + 2*q^38 + q^50 + 2*q^59 + 2*q^83 + 2*q^86 + q^98 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A255317, A255318, A257398, A257399, A257403.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^48] QPochhammer[ -x^16, x^16], {x, 0, n}];

{a(n) = my(A, p, e); if( n<0 || n%8 == 5, 0, A = factor(3*n + 2); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p+e==3, p%8 > 4, 1-e%2, e+1)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^16 + A) * eta(x^96 + A)), n))};

Expansion of q^(-2/3) * eta(q^6)^5 * eta(q^32) * eta(q^48)^2 / (eta(q^3)^2 * eta(q^12)^2 * eta(q^16) * eta(q^96)) in powers of q.
Euler transform of a period 96 sequence.
a(n) = A257403(3*n + 2) unless n == 5 (mod 8).
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.
a(4*N) = A257399(n). a(8*n+3) = 2*A255318(n). a(16*n) = A257398(n). a(16*n+12) = 2*A255317(n).

cp's OEIS Frontend

A257399 Expansion of phi(x^3) * phi(-x^12) / chi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A257403 Multiplicative with a(2) = 1, a(2^e) = 0 if e>1, a(3^e) = 0^e, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A256505 Expansion of phi(x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula