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A028641
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Expansion of (theta_3(z)*theta_3(19z)+theta_2(z)*theta_2(19z)).
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1, 2, 0, 0, 2, 4, 0, 4, 0, 2, 0, 4, 0, 0, 0, 0, 2, 4, 0, 2, 4, 0, 0, 4, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 4, 0, 6, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 4, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 2, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 6, 4, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Theta series of quadratic form with Gram matrix [ 2, 1; 1, 10 ].
Expansion of phi(q)* phi(q^19) +4*q^5* psi(q^2)* psi(q^38) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos Feb 27 2007
Moebius transform is period 19 sequence [ 2, -2, -2, 2, 2, 2, 2, -2, 2, -2, 2, -2, -2, -2, -2, 2, 2, -2, 0, ...]. - Michael Somos Feb 27 2007
a(n)=2*b(n) where b(n) is multiplicative and b(19^e) = 1, b(p^e) = e+1 if kronecker(-19, p) = 1, b(p^e) = (1+(-1)^e)/2 if kronecker(-19, p) = -1. - Michael Somos Feb 27 2007
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PROG
| (PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-19, d))*2)} /* Michael Somos Feb 27 2007 */
(PARI) {a(n)=if(n<1, n==0, qfrep([2, 1; 1, 10], n, 1)[n]*2)} /* Michael Somos Feb 27 2007 */
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CROSSREFS
| Sequence in context: A020474 A135589 A158122 * A141416 A176787 A127862
Adjacent sequences: A028638 A028639 A028640 * A028642 A028643 A028644
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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