A111317 Let f(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is f(q^2,q^3) / f(q,q^3).

1, 1, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, -1, 0, 2, -1, -1, 2, -1, -2, 3, 1, -3, 2, 1, -4, 2, 3, -4, 1, 4, -5, 0, 6, -5, -2, 7, -5, -4, 10, -3, -7, 10, -2, -10, 11, 1, -13, 11, 4, -16, 11, 9, -19, 8, 12, -22, 7, 19, -24, 2, 24, -26, -3, 32, -25, -10, 37, -25, -18, 45, -21, -29, 49, -17, -39, 56, -8, -51, 58, 0, -65, 61, 14, -78, 59, 27, -92

Offset: 0

Author

N. J. A. Sloane, Nov 09 2005

Keywords

• sign
• look

Comments

Convolution inverse of A111165.

Examples

			From _Peter Bala_, Dec 2012: (Start)
F(1/10) = Sum_{n>=0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(9 + 1/(1 + 1/(99 + 1/(1 + 1/(999 + 1/(1 + ...)))))).
F(-1/10) = Sum_{n>=0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(9 + 1/(101 + 1/(999 + 1/(1001 + ...))))).
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Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 25, Equation (39).

Crossrefs

Cf. A005169, A143951.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0, 1, -1][irem(d, 3)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*{0, 1, -1}[[Mod[d, 3]+1]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=0, n\3, (1 - x^(3*k+2)) / (1 - x^(3*k+1)), 1 + x * O(x^n)), n))} /* Michael Somos, Dec 23 2007 */

PARI

{a(n)=polcoeff(exp(sum(m=1,n+1,1/(1+x^m+x^(2*m)+x*O(x^n))*x^m/m)),n)} \\ Paul D. Hanna, Jan 23 2010

Sage

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(-1, -1)
a = EulerTransform(b)
print([a(n) for n in range(88)]) # Peter Luschny, Nov 17 2022

Formula

Euler transform of period 3 sequence [ 1, -1, 0, ...]. - Michael Somos, Dec 23 2007

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

G.f.: exp( Sum_{n>=1} 1/(1 + x^n + x^(2n)) * x^n/n ). - Paul D. Hanna, Jan 23 2010

From Peter Bala, Dec 2012: (Start)

Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 2, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-1 + 1/(1 + 1/(n^2-1 + 1/(1 + 1/(n^3-1 + 1/(1 + ...)))))).

For n >= 2, F(-1/n) has the simple continued fraction expansion

1/(1 + 1/(n-1 + 1/(n^2+1 + 1/(n^3-1 + ...)))). Examples are given below. Cf. A005169 and A143951.

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