OFFSET
0,1
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 89, Equation (1.2), page 100, Equation (5.3)
G. E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See page 3, Equation (1.5)
FORMULA
Expansion of Sum_{k>=0} x^k / (Product_{j=0..k} ( 1 + x^(2*k + 1) ) ) in powers of x^2. - Michael Somos, Nov 04 2013
a(n) = b(3*n + 1) where b() is multiplicative with b(p^(2*e)) = -(-1)^e if p = 2, b(p^(2*e)) = (-1)^e if p = 5 (mod 6), b(p^(2*e)) = 1 if p = 1 (mod 6), and b(p^(2*e-1)) = b(3^e) = 0 if e>0. - Michael Somos, Jul 19 2013
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0.
G.f.: Sum_{k>=0} (-1)^k * x^(3*k^2 + 2*k) * ( 1 + x^(2*k + 1) ).
G.f.: 1/(1 - x*(1-x)/(1 - x^2*(1-x^2)/(1 - x^3*(1-x^3)/(1 - x^4*(1-x^4)/(1 - ...))))), a continued fraction. - Paul D. Hanna, Jul 18 2013
abs(a(n)) = A089801(n). - Michael Somos, Jun 30 2015
G.f.: 1 + x*(1-x) + x^2*(1-x)*(1-x^3) + x^3*(1-x)*(1-x^3)*(1-x^5) + ... . - Michael Somos, Aug 03 2017
EXAMPLE
G.f. = 1 + x - x^5 - x^8 + x^16 + x^21 - x^33 - x^40 + x^56 + x^65 - x^85 - x^96 + ...
G.f. = q + q^4 - q^16 - q^25 + q^49 + q^64 - q^100 - q^121 + q^169 + q^196 + ...
MATHEMATICA
a[ n_] := With[ {m = Sqrt[3 n + 1]}, If[ IntegerQ @ m, (-1)^Quotient[ m, 3], 0]]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ (-1)^k x^(3 k^2 + 2 k) (1 + x^(2 k + 1)), {k, 0, n}], {x, 0, n}]; (* Michael Somos, Nov 04 2013 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k QPochhammer[ x, x^2, k], {k, 0, n}], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k / QPochhammer[ -x, x^2, k + 1], {k, 0, 2 n}], {x, 0, 2 n}]; (* Michael Somos, Jun 30 2015 *)
PROG
(PARI) {a(n) = my(m); if( issquare( 3*n + 1, &m), (-1)^(m \ 3) )};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( e%2, 0, p==2, -(-1)^(e/2), p == 3, 0, p%6 == 1, 1, (-1)^(e/2))))}; /* Michael Somos, Jul 19 2013 */
(PARI) /* Continued Fraction: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 18 2013
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved