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A143064 Expansion of a Ramanujan false theta series variation of A089801 in powers of x. 9
1, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.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.

a(8*n) = A143062(n). Convolution of A010054 with A143065. - Michael Somos, Jul 19 2013

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

Cf. A089801, A010054, A143062, A143065.

Column m=0 of A185646.

Sequence in context: A089802 A089801 A290739 * A185124 A185125 A327580

Adjacent sequences:  A143061 A143062 A143063 * A143065 A143066 A143067

KEYWORD

sign

AUTHOR

Michael Somos, Jul 21 2008

STATUS

approved

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Last modified February 21 20:35 EST 2020. Contains 332111 sequences. (Running on oeis4.)