A256574 -id:A256574 - cp's OEIS frontend

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).

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

A255320 Expansion of chi(-x) * psi(x^3) * psi(x^48) in powers of x where chi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Feb 21 2015

sign

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

			G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^48 - x^49 + ...
G.f. = q^19 - q^22 - q^34 + q^43 + q^67 - q^82 - q^118 + q^139 + q^163 + ...

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. A227395, A255317, A255318, A255319, A256574.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[2, 0, x^(24)] / (4 x^(51/8)), {x, 0, n}];

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

{a(n) = my(A, p, e); if( n<0 || n%8 == 2, 0, A = factor(3*n + 19); 1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(p+e==3), p%8 > 4, 1-e%2, e+1)))}; /* Michael Somos, Apr 24 2015 */

Expansion of q^(-19/3) * eta(q) * eta(q^6)^2 * eta(q^96)^2 / (eta(q^2) * eta(q^3) * eta(q^48)) in powers of q.
Euler transform of a period 96 sequence.
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = a(16*n + 9) = a(16*n + 13) = 0.
-2 * a(n) = A227395(3*n + 19). a(8*n) = A255317(n). a(16*n + 1) = -A255318(n). a(16*n + 5) = -A255319(n).
a(n) = (-1)^n * A256574(n). - Michael Somos, Apr 24 2015

A263767 Expansion of phi(-x) * psi(-x^8) * chi(x^24) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 25 2015

sign

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

			G.f. = 1 - 2*x + 2*x^4 - x^8 - 2*x^12 + 2*x^16 + 2*x^17 - 2*x^24 - 2*x^25 + ...

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. A256574.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^24, x^48] EllipticTheta[ 2, Pi/4, x^4] EllipticTheta[ 4, 0, x] / (2^(1/2) x), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A) * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^2 + A) * eta(x^16 + A) * eta(x^24 + A) * eta(x^96 + A)), n))};

q='q+O('q^99); Vec(eta(q)^2*eta(q^8)*eta(q^32)*eta(q^48)^2/(eta(q^2)*eta(q^16)* eta(q^24)*eta(q^96))) \\ Altug Alkan, Jul 31 2018

Expansion of eta(q)^2 * eta(q^8) * eta(q^32) * eta(q^48)^2 / (eta(q^2) * eta(q^16) * eta(q^24) * eta(q^96)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 10368^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A256574.
a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.

cp's OEIS Frontend

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).

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A255320 Expansion of chi(-x) * psi(x^3) * psi(x^48) in powers of x where chi(), psi() are Ramanujan theta functions.

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A263767 Expansion of phi(-x) * psi(-x^8) * chi(x^24) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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