A143065 Expansion of quotient of a Ramanujan false theta series by the theta series of triangular numbers in powers of x.

1, 0, 0, -1, 1, -2, 2, -3, 4, -5, 6, -8, 11, -13, 16, -21, 27, -32, 39, -49, 61, -73, 87, -107, 131, -155, 184, -223, 267, -315, 372, -443, 526, -617, 722, -852, 1002, -1167, 1359, -1590, 1854, -2148, 2488, -2888, 3346, -3859, 4444, -5128, 5909, -6779, 7773

Offset

0,6

References

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.

Links

Table of n, a(n) for n=0..50.

Formula

G.f.: ( 1 + x - x^5 - x^8 + x^16 + x^21 - ... ) / ( 1 + x + x^3 + x^6 + x^10 + x^15 + ... ). [Ramanujan]

G.f.: 1 - x^3 * (1 - x) / (1 - x^2)^2 + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2)^2 * (1 - x^4)^2) - ... [Ramanujan]

Convolution with A010054 is A143064.

Example

G.f. = 1 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + 6*x^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {x^2}, x^2, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, if( issquare( 3*k + 1, &m), (-1)^(m \ 3) * x^k ), A) / sum(k=0, (sqrtint(8*n + 1) - 1) \ 2, x^((k^2 + k) / 2), A), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n+1) - 1, (-1)^k * x^(k^2 + 2*k) * prod(j=1, k, (1 - x^(2*j - 1)) / (1 - x^(2*j))^2, 1 + O(x^(n + 1 - k^2 - 2*k)))), n))};

Crossrefs

Cf. A010054, A143064.

Sequence in context: A130082 A241377 A266750 * A192660 A173692 A316079

Adjacent sequences: A143062 A143063 A143064 * A143066 A143067 A143068

Keyword

sign

Author

Michael Somos, Jul 21 2008

Status

approved