A282937 a(n) = A000728(5*n).

1, -6, 9, 10, -30, 1, 5, 51, 10, -100, 20, -55, 109, 110, -130, -1, -110, 160, 10, -230, 100, 15, 191, 120, -230, -100, -89, 160, 90, -340, 120, 5, 300, 200, -260, -1, -275, 240, -100, -270, 119, -165, 260, 410, -200, -40, 20, 200, -110, -500, 180, -54, 140

Offset

0,2

Comments

This generalized function is related to two following identities; R(q^5) - q - q^2/R(q^5) = (q; q)_{infinity}/(q^25; q^25)_{infinity}, R^5(q^5) - 11*q^5 - q^10/R^5(q^5) = ((q^5; q^5)_{infinity}/(q^25; q^25)_{infinity})^6, where R(q) is the Rogers-Ramanujan continued function and (q; q)_n is the q-Pochhammer symbol. See the reference.

References

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 185.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Formula

G.f.: Product_{n>=1} (1 - q^n)^6/(1 - q^(5*n)).

a(n) = (-1)^j mod 5 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 5.

Sum_{k=0..n} a(k)*A182821(n-k) = 0 for n > 0. - Seiichi Manyama, Feb 28 2017

G.f.: exp( Sum_{n>=1} -sigma(5*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017

a(n) = -(1/n)*Sum_{k=1..n} sigma(5*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

Example

G.f.: 1 - 6*q + 9*q^2 + 10*q^3 - 30*q^4 + q^5 + 5*q^6 + 51*q^7 + ...

Mathematica

CoefficientList[Series[Product[(1 - x^j)^6/(1 - x^(5*j)), {j, 1, 62}], {x, 0, 60}], x] (* G. C. Greubel, Nov 18 2018 *)

Prog

(PARI) m=60; x='x+O('x^m); Vec(prod(j=1, m+2, (1 - x^j)^6/(1 - x^(5*j)))) \\ G. C. Greubel, Nov 18 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
prec = 60
x = R.gen().O(prec)
s = prod((1 - x^j)^6/(1 - x^(5*j)) for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Crossrefs

Cf. A000728, A182821 (Product_{n>=1} (1 - q^(5*n))/(1 - q^n)^6), A282941.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), this sequence (k=5), A282942 (k=7).

Sequence in context: A209941 A000729 A280666 * A106248 A132725 A338596

Adjacent sequences: A282934 A282935 A282936 * A282938 A282939 A282940

Keyword

sign

Author

Seiichi Manyama, Feb 25 2017

Status

approved