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Expansion of quotient of a Ramanujan false theta series by the theta series of triangular numbers in powers of x.
1

%I #7 Nov 08 2015 09:56:32

%S 1,0,0,-1,1,-2,2,-3,4,-5,6,-8,11,-13,16,-21,27,-32,39,-49,61,-73,87,

%T -107,131,-155,184,-223,267,-315,372,-443,526,-617,722,-852,1002,

%U -1167,1359,-1590,1854,-2148,2488,-2888,3346,-3859,4444,-5128,5909,-6779,7773

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

%D S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.

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

%F 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]

%F Convolution with A010054 is A143064.

%e 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 + ...

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

%o (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))};

%o (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))};

%Y Cf. A010054, A143064.

%K sign

%O 0,6

%A _Michael Somos_, Jul 21 2008