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G.f.: Product_{n>0} (1 - x^n)^3 * (1 - x^(2*n - 1))^2 = Sum_{n in Z} (6*n + 1) * x^(n*(3*n + 1)/2).
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%I #34 Jan 04 2021 06:21:40

%S 1,-5,7,0,0,-11,0,13,0,0,0,0,-17,0,0,19,0,0,0,0,0,0,-23,0,0,0,25,0,0,

%T 0,0,0,0,0,0,-29,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,-35,0,0,0,0,0,37,0,0,

%U 0,0,0,0,0,0,0,0,0,0,-41,0,0,0,0,0,0,43,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-47,0,0,0,0,0,0

%N G.f.: Product_{n>0} (1 - x^n)^3 * (1 - x^(2*n - 1))^2 = Sum_{n in Z} (6*n + 1) * x^(n*(3*n + 1)/2).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 306.

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.6); p. 84, Eq. (32.63).

%D S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)

%H Seiichi Manyama, <a href="/A080332/b080332.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F G.f.: theta_4(x)^2 * (Sum_{n in Z} (-1)^n * x^(n*(3*n + 1)/2)).

%F Expansion of f(-x)^2 * phi(x) = f(-x^2) * phi(-x^2)^2 in powers of x^2 where phi(), f() are Ramanujan theta functions. - _Michael Somos_, Feb 18 2003

%F Expansion of q^(-1/24) * eta(q)^5 / eta(q^2)^2 in powers of q.

%F Euler transform of period 2 sequence [-5, -3, ...]. - _Michael Somos_, Sep 09 2007

%F a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2* p^(e/2) if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 * (-p)^(e/2) if p == 5 (mod 6). - _Michael Somos_, May 26 2005

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113277. - _Michael Somos_, Feb 18 2003

%F a(5*n + 3) = a(5*n + 4) = a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0. a(25*n + 1) = -5 * a(n). - _Michael Somos_, Feb 18 2003

%e G.f. = 1 - 5*x + 7*x^2 - 11*x^5 + 13*x^7 - 17*x^12 + 19*x^15 - 23*x^22 + ...

%e G.f. = q - 5*q^25 + 7*q^49 - 11*q^121 + 13*q^169 - 17*q^289 + 19*q^361 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^2]^2, {x, 0, n}]; (* _Michael Somos_, Mar 11 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* _Michael Somos_, Mar 11 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 3, 0, x], {x, 0, 2 n}]; (* _Michael Somos_, Mar 11 2015 *)

%t a[ n_] := With[{m = Sqrt[24 n + 1]}, If[ IntegerQ[ m], m KroneckerSymbol[ -3, m], 0]]; (* _Michael Somos_, Mar 11 2015 *)

%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( (p<5) || (e%2), 0, if( p%6 == 1, p, -p)^(e\2))))}; /* _Michael Somos_, May 26 2005 */

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^2 + A)^2, n))};

%o (PARI) {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -3, n), 0)};

%Y Cf. A010815, A113277.

%K sign,easy

%O 0,2

%A _Michael Somos_, Feb 18 2003

%E Definition changed by _N. J. A. Sloane_, Aug 14 2007