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

Author

Seiichi Manyama, Feb 25 2017

Keywords

• sign

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.

Examples

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

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

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).

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018

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 *)

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

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

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