A226556 Expansion of f(x, -x^4) / f(-x^2, x^3) in powers of x where f(,) is Ramanujan's general theta function.

1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 2, 3, 2, 0, -2, -4, -4, -3, -1, 3, 6, 7, 5, 0, -5, -9, -10, -7, -1, 7, 14, 16, 11, 1, -11, -20, -22, -16, -2, 15, 29, 33, 23, 2, -23, -41, -45, -32, -4, 30, 57, 64, 45, 4, -43, -78, -86, -60, -7, 57, 107, 119, 83, 8, -79

Offset

0,11

Comments

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

Given the g.f. A(x), then S(q) := q^(1/5) * A(q) notation is used by Berndt.

References

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 57.

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 9.

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

O. Marichev and M. Trott, After 100 Years, Ramanujan Gap Filled

Formula

Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (1 + u*v^3) - 3 * u^2*v^2.

Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^5 * (1 - 3*v + 4*v^2 - 2*v^3 + v^4) - v * (1 + 2*v + 4*v^2 + 3*v^3 + v^4).

Euler transform of period 20 sequence [1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -1, -1, 0, 1, 0, ...].

G.f.: (Sum_{k in Z} (-1)^k * (-x)^((5*k + 3)*k / 2)) / (Sum_{k in Z} (-1)^k * (-x)^((5*k + 1)*k / 2)).

G.f.: 1 / (1 - x / (1 + x^2 / (1 - x^3 / ...))). [continued fraction]

a(n) = (-1)^n * A007325(n).

G.f.: 1/G(0), where G(k)= 1 + (-x)^(k+1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013

Example

G.f. = 1 - x + x^2 - x^4 + x^5 - x^6 + x^7 - x^9 + 2*x^10 - 3*x^11 + 2*x^12 + ...
G.f. = q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + 2*q^61 + ...

Mathematica

a[ n_] := SeriesCoefficient[ ContinuedFractionK[ (-q)^k, 1, {k, 0, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q^5] QPochhammer[ q^4, -q^5] / (QPochhammer[ q^2, -q^5] QPochhammer[-q^3, -q^5]), {q, 0, n}];

Prog

(PARI) {a(n) = my(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + 3*n) / 2), x * O(x^n)) / sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + n) / 2), x * O(x^n)), n))};

Crossrefs

Cf. A007325.

Sequence in context: A002120 A021435 A334358 * A007325 A247920 A269735

Adjacent sequences: A226553 A226554 A226555 * A226557 A226558 A226559

Keyword

sign

Author

Michael Somos, Jun 10 2013

Status

approved