A258832 -id:A258832 - cp's OEIS frontend

A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

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

Offset: 0

Michael Somos, Jul 30 2006

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

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^5 + q^17 + q^29 + q^41 + q^53 + 2*q^65 + q^89 + q^101 + q^113 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
K. Saito, "Extended Affine Root Systems. V. Elliptic Eta-Products and Their Dirichlet Series", Proceedings on Moonshine and related topics (Montreal, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001. MR1881609 (2003d:11066) See page 215.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A098098, A121363, A258210, A258291, A258832.

a[ n_] := If[ n < 0, 0, Sum[ I^d, {d, Divisors[12 n + 5]}] / (2 I)]; (* Michael Somos, Jul 25 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(3/8) QPochhammer[ x^6]^3 / (QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)]), {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := Length @ FindInstance[ 24 n + 10 == (6 j + 3)^2 + (6 k + 1)^2 && j >= 0, {j, k}, Integers, 10^9]; (* Michael Somos, Jul 02 2015 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 0, 0, Sum[ Boole[ Mod[d, 4] == 1] - Boole[ Mod[d, 4] == 3], {d, Divisors[12 n + 5]}] / 2]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

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

{a(n) = if( n<0, 0, n = 12*n + 5; sumdiv(n, d, (d%4==1) - (d%4==3)) / 2)};

Expansion of f(-x^3) * f(-x^6) / chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/12) * eta(q^2) * eta(q^3) * eta(q^6) / eta(q) in powers of q.
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258210.
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)).
-2 * a(n) = A121363(3*n + 1).
Convolution square is A098098.
a(n) = (-1)^n * A258832(n) = A052343(3*n + 1). -a(n) = A258291(3*n + 1). 2 * a(n) = A008441(3*n + 1). - Michael Somos, Jul 02 2015
From Peter Bala, Jan 07 2021: (Start)
G.f. A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(12*n + 5)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(12*n + 5)). Cf. A033761. (End)

A258831 Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -4, 5, -8, 7, -8, 9, -10, 14, -12, 16, -14, 15, -20, 17, -18, 19, -24, 26, -22, 23, -28, 25, -32, 32, -28, 29, -30, 38, -32, 33, -40, 40, -44, 42, -38, 39, -40, 57, -42, 43, -44, 45, -62, 47, -56, 49, -56, 62, -52, 53, -60, 64, -68, 64, -58, 59, -60

Offset: 0

Michael Somos, Jun 11 2015

sign

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

			G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 5*x^4 - 8*x^5 + 7*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q^5 - 2*q^11 + 3*q^17 - 4*q^23 + 5*q^29 - 8*q^35 + 7*q^41 - 8*q^47 + ...

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. A098098, A121613, A208435, A208457, A258832.

List([0..70], n -> (-1)^n*Sigma(6*n+5)/6); # Muniru A Asiru, Jan 30 2018

[(-1)^n*SumOfDivisors(6*n+5)/6: n in [0..70]]; // Vincenzo Librandi, Jan 30 2018

with(numtheory):
seq((-1)^(n-1)*sigma(6*n - 1)/6, n=1..10^3); # Muniru A Asiru, Jan 30 2018

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 6 n + 5] / 6];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3])^2, {x, 0, n}];
Table[(-1)^n DivisorSigma[1, 6 n + 5] / 6, {n, 0, 60}] (* Vincenzo Librandi, Jan 30 2018 *)

{a(n) = if(n<0, 0, (-1)^n*sigma(6*n+5)/6)};

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

Expansion of (f(-x^6)^2 * chi(x^3) / chi(x))^2 in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/6) * (eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [-2, 2, 0, 0, -2, -4, -2, 0, 0, 2, -2, -4, ...].
a(n) = (-1)^n * A098098(n) = A208435(2*n + 1) = A208457(2*n + 1). 6 * a(n) = A121613(3*n + 2).
Convolution square of A258832.

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A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

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A258831 Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

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