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A127693
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Expansion of psi(x^2)+x*psi(x^10) in powers of x where psi() is a Ramanujan theta function.
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1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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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REFERENCES
| R. Blecksmith; J. Brillhart; I. Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
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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
| Euler transform of period 20 sequence [ 1, 0, -1, 0, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, 0, -1, 0, 1, -1, ...].
a(n)=b(4n+1) where b(n) is multiplicative and b(2^e)=0^e, b(5^e)=1, else b(p^e)=(1+(-1)^e)/2.
a(9n+2)=a(5n+1)=a(n), a(5n+3)=a(5n+4)=a(6n+3)=a(6n+4)=a(9n+5)=a(9n+8)=0.
G.f.: Sum_{k>0} x^(k(k-1)) +x^(5k(k-1)+1) = Product_{k>0} (1-x^(10k)) (1+x^(10k-1)) (1+x^(10k-2)) (1-x^(10k-3)) (1+x^(10k-4)) (1+x^(10k-6)) (1-x^(10k-7)) (1+x^(10k-8)) (1+x^(10k-9)).
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PROG
| (PARI) {a(n)=issquare(4*n+1)+issquare(20*n+5)}
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CROSSREFS
| Cf. A010054(n)=a(2n). A089806(n)=a(3n). A080995(n)=a(6n).
Sequence in context: A157022 A016362 A016415 * A129405 A127001 A068431
Adjacent sequences: A127690 A127691 A127692 * A127694 A127695 A127696
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Jan 19 2007
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