A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

1, -5, 5, 10, -15, -1, -30, 50, 65, -95, -1, -170, 220, 300, -380, 0, -635, 820, 1025, -1310, 0, -2045, 2525, 3140, -3845, 2, -5780, 7070, 8565, -10405, -1, -15130, 18125, 21760, -25960, 0, -36820, 43780, 51785, -61290, 0, -85170, 100030, 117500, -137550, 0

Offset

0,2

Comments

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

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

a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.

Expansion of q^(5/6) * eta(q)^5 / eta(q^5)^5 in powers of q. - Michael Somos, Apr 29 2017

Expansion of f(-x)^5 / f(-x^5)^5 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Apr 29 2017

Euler transform of period 5 sequence [-5, -5, -5, -5, 0, ...]. - Michael Somos, Apr 29 2017

Example

G.f. = 1 - 5*x + 5*x^2 + 10*x^3 - 15*x^4 - x^5 - 30*x^6 + 50*x^7 + 65*x^8 - 95*x^9 + ...
G.f. = q^-5 - 5*q + 5*q^7 + 10*q^13 - 15*q^19 - q^25 - 30*q^31 + 50*q^37 + 65*q^43 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^5]^5, {x, 0, n}]; (* Michael Somos, Apr 29 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^5, n))}; /* Michael Somos, Apr 29 2017 */

Crossrefs

(Product_{k>0} (1 - x^k) / (1 - x^(m*k)))^m: A022597 (m=2), A199659 (m=3), A112143 (m=4), this sequence (m=5).

Cf. A285928.

Sequence in context: A029842 A112436 A309457 * A109064 A138506 A000728

Adjacent sequences: A285929 A285930 A285931 * A285933 A285934 A285935

Keyword

sign

Author

Seiichi Manyama, Apr 29 2017

Status

approved