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 A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ]. 1

%I

%S 1,0,0,2,2,2,0,0,0,2,0,0,4,0,0,2,2,0,0,0,4,0,0,2,0,2,0,2,0,0,0,2,0,2,

%T 0,0,6,2,0,0,0,0,0,0,2,4,0,0,4,0,0,0,0,0,0,2,0,0,0,2,8,0,0,0,2,0,0,2,

%U 0,2,0,2,0,0,0,4,0,0,0,0,4,4,0,0,0,0,0,0,0,2,0,0,4,2,0,0,0,2,0,2,6,0,0,0,0

%N Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].

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

%H John Cannon, <a href="/A028953/b028953.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">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 Expansion of phi(q) * phi(q^11) - 2*q * f(-q^2) * f(-q^22) = phi(q^3) * phi(q^33) + 2*q^4 * chi(q) * psi(-q^3) * chi(q^11) * psi(-q^33) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions - Michael Somos and Alex Berkovich, Jun 24 2011

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44^(1/2) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos, Jun 24 2011

%F G.f.: Sum_{n,m} x ^ (3*n*n + 2*n*m + 4*m*m).

%F a(4*n + 2) = a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0. - Michael Somos, Feb 23 2012

%e 1 + 2*q^3 + 2*q^4 + 2*q^5 + 2*q^9 + 4*q^12 + 2*q^15 + 2*q^16 + 4*q^20 + 2*q^23 + 2*q^25 + 2*q^27 + 2*q^31 + 2*q^33 + 6*q^36 + 2*q^37 + 2*q^44 + 4*q^45 + ...

%o (PARI) {a(n) = if( n<1, n==0, qfrep([3, 1; 1, 4], n)[n] * 2)} /* Michael Somos, Jun 24 2011 */

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\11), 2 * x^(11*k^2), 1 + A) - 2 * x * eta(x^2 + A) * eta(x^22 + A), n))} /* Michael Somos, Jun 24 2011 */

%K nonn

%O 0,4

%A _N. J. A. Sloane_.

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Last modified May 21 08:28 EDT 2013. Contains 225478 sequences.