A113429 Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset

0,1

Comments

For the g.f. identity see the Hardy-Wright reference, Theorem 355 on p. 284. - Wolfdieter Lang, Oct 28 2016

References

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

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

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Formula

Euler transform of period 5 sequence [-1, 0, 0, -1, -1, ...].

|a(n)| is the characteristic function of A085787.

G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k-1)) * (1 - x^(5*k-4)) = Sum_{k in Z} (-1)^k * x^((5*k^2+3*k)/2).

f(a, b) = Sum_{k in Z} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's general theta function.

G.f.: Sum_{n>=0} (x^(n*(n+1)) * Product_{k>=n+1} (1-x^k)). - Joerg Arndt, Apr 07 2011

From Wolfdieter Lang, Oct 30 2016: (Start)

a(n) = (-1)^k if n = b(2*k) for k >= 0, a(n) = (-1)^k if n = b(2*k-1), for k >= 1, and a(n) = 0 otherwise, where b(n) = A085787(n). See the second formula.

G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+3)/2)*(1-x^(2*n+1)). See the Hardy reference, p. 93, G_1(x,x) from eq. (6.11.1) with C_n(x,x) = 1.

(End)

G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n-3)/2)*(1-x^(4*(2*n+1)). Reordered G_1(x,x) from the preceding formula. This is G_4(x,x) from Hardy, p. 93, eq. (6.11.1) with C_n(x,x) = 1. Note that Hardy uses only G_0, G_1 and G_2. - Wolfdieter Lang, Nov 01 2016

a(n) = -(1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Example

G.f. = 1 - x - x^4 + x^7 + x^13 - x^18 - x^27 + x^34 + x^46 - x^55 - x^70 + ...
G.f. = q^9 - q^49 - q^169 + q^289 + q^529 - q^729 - q^1089 + q^1369 + q^1849 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^5] QPochhammer[ x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)
a[ n_] := Module[{m = 40 n + 9, k}, If[IntegerQ[k = Sqrt[m]], If[Mod[k, 10] == 7, k = -k]; (-1)^Quotient[k, 10], 0]]; (* Michael Somos, Jun 26 2017 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k*[1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};
(PARI) {a(n) = my(m, k); if( n<0, 0, issquare(m = 40*n + 9, &k), if( k%10==7, k=-k); (-1)^(k\10), 0)}; /* Michael Somos, Oct 29 2016 */

Crossrefs

Cf. A085787, A113428.

Sequence in context: A358680 A186447 A118009 * A133100 A216230 A077606

Adjacent sequences: A113426 A113427 A113428 * A113430 A113431 A113432

Keyword

sign,easy

Author

Michael Somos, Oct 31 2005

Status

approved