A124242 Expansion of a parametrization of Ramanujan's continued fraction.

1, -1, 1, 1, -2, 0, 2, -2, -1, 4, -1, -4, 4, 1, -6, 3, 6, -7, -3, 10, -4, -10, 12, 6, -18, 5, 18, -20, -8, 30, -10, -29, 31, 12, -46, 17, 44, -47, -20, 68, -23, -66, 72, 31, -104, 33, 98, -107, -44, 156, -51, -144, 154, 61, -220, 75, 206, -220, -90, 310, -104, -290, 312, 131, -442, 143, 408, -437, -178, 618, -202

Offset

0,5

Comments

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

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53

Links

Seiichi Manyama, 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

Formula

Euler transform of period 10 sequence [ -1, 1, 2, -1, -2, -1, 2, 1, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - (2-u) * (2 - (2-u) * (2-v)).

Given g.f. A(x) =: k, then B(x) = (1-k) * (k / (2-k))^2, B(x^2) = (1-k)^2 * ((2-k) / k) where B(x) is the g.f. for A078905.

Expansion of f(-x^5, -x^10)^3 / (f(x, x^4) * f(-x^3, -x^7)^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 06 2016

G.f.: Product_{k>0} ((1 - x^(10k-5)) / ((1 - x^(10k-3)) * (1 - x^(10k-7))))^2 * (1 - x^(10k-1)) * (1 - x^(10k-4)) * (1 - x^(10k-6)) * (1 - x^(10k-9) / ((1-x^(10k-2)) * (1-x^(10k-8))).

-a(n) = A112274(n) unless n = 0.

G.f.: 1 - r(q) * r(q^2)^2 where r() is the Rogers-Ramanujan continued fraction. - Seiichi Manyama, Apr 18 2017

Example

G.f. = 1 - x + x^2 + x^3 - 2*x^4 + 2*x^6 - 2*x^7  - x^8 + 4*x^9 - x^10 - 4*x^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 1, -1, -2, 1, 2, 1, -2, -1, 1, 0}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + A)^[0, 1, -1, -2, 1, 2, 1, -2, -1, 1][k%10+1]), n))};

Crossrefs

Cf. A078905, A112274, A112803, A285348.

Sequence in context: A117963 A321594 A112803 * A112274 A336891 A181391

Adjacent sequences: A124239 A124240 A124241 * A124243 A124244 A124245

Keyword

sign

Author

Michael Somos, Oct 27 2006

Status

approved