A230534 -id:A230534 - cp's OEIS frontend

A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, -1, 2, 1, -2, -2, 2, 4, -4, -5, 6, 6, -7, -9, 10, 13, -15, -17, 20, 21, -25, -28, 32, 39, -43, -49, 56, 60, -69, -78, 86, 101, -112, -125, 142, 153, -172, -192, 212, 241, -266, -295, 328, 357, -397, -438, 482, 540, -592, -652, 720, 781, -862, -946

Offset: 0

Michael Somos, Jun 10 2013

sign

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

			G.f. = 1 - x - x^2 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 + 4*x^8 - 4*x^9 + ...
G.f. = q^3 - q^7 - q^11 + 2*q^15 + q^19 - 2*q^23 - 2*q^27 + 2*q^31 + 4*q^35 + ...

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. A092869, A230534.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, 1, 1, -1, -2, -1, 1, 1}[[ Mod[k, 8] + 1]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, 1, -1, -2, -1, 1, 1][k%8 + 1]), n))};

Expansion of (f(-x^1, -x^7) / f(-x^3, -x^5)) * (psi(-x^2) / phi(-x^4)) in powers of x where psi(), phi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [-1, -1, 1, 2, 1, -1, -1, 0, ...].
Given g.f. A(x) then B(q) = q^3 * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v - u^2)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (2 + v^2 - u^2*v).
a(n) = -A092869(2*n + 1) = A230534(2*n + 1).

A230535 Expansion of q * (f(-q, -q^7) / f(-q^3, -q^5))^2 in powers of q where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, -2, 1, 2, -4, 4, -1, -6, 11, -8, -1, 12, -20, 16, 2, -22, 34, -30, 1, 40, -64, 52, -2, -68, 113, -88, -2, 112, -180, 144, 2, -182, 284, -228, 4, 286, -452, 356, -4, -440, 698, -544, -5, 668, -1044, 816, 6, -996, 1545, -1210, 6, 1464, -2276, 1768, -7, -2128

Offset: 1

Michael Somos, Oct 22 2013

sign

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

			G.f. = q - 2*q^2 + q^3 + 2*q^4 - 4*q^5 + 4*q^6 - q^7 - 6*q^8 + 11*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A092869, A224216, A230534.

a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 8, k], {k, n}]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2] / (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^8] QPochhammer[ q^7, q^8] / (QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8]))^2, {q, 0, n}];

{a(n) = local(A, A2); if( n<2, n==1, n--; A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( (2 * A^2 * A2^2 / (A^2 + A2))^2, n))};

Euler transform of period 8 sequence [ -2, 0, 2, 0, 2, 0, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - u^2) * (1 - v)  - 4*u*v.
G.f.: x * (Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))))^2.
a(2*n) = -2 * A224216(n). a(2*n + 1) = A230534(n).
Convolution square of A092869.
Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)). - Simon Plouffe, Mar 02 2021

cp's OEIS Frontend

A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

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