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Search: id:A112274
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| A112274 |
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Expansion of parametrization of Ramanujan's continued fraction. |
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+0
3
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| 1, -1, -1, 2, 0, -2, 2, 1, -4, 1, 4, -4, -1, 6, -3, -6, 7, 3, -10, 4, 10, -12, -6, 18, -5, -18, 20, 8, -30, 10, 29, -31, -12, 46, -17, -44, 47, 20, -68, 23, 66, -72, -31, 104, -33, -98, 107, 44, -156, 51, 144, -154, -61, 220, -75, -206, 220, 90, -310, 104, 290, -312, -131, 442, -143, -408, 437, 178, -618, 202, 567
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Cumulative sums are: 1, 0, -1, 1, 1, -1, 1, 2, -2, -1, 3, -1, -2, 4, 1, -5, ...-5, 2, 5, -5, -1, 9, -3, -9, 9, 4, -14, 6, 14, -16, -6, 23. Conjecture: limit_[n goes to infinity] (cumulative sum of A112274)/n = 0. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Sep 01 2005
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REFERENCES
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53
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FORMULA
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Euler transform of period 10 sequence [ -1, -1, 1, 1, 0, 1, 1, -1, -1, 0, ...].
G.f.: x(f(-x^2, -x^8)f(-x, -x^9))/(f(-x^4, -x^6)f(-x^3, -x^7)) where f(a, b) is Ramanujan's two variable theta function.
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=(u+v)^2+v*(u^2-1).
G.f.: Product_{k>0} (1-x^(10k-1))(1-x^(10k-2))(1-x^(10k-8))(1-x^(10k-9))/((1-x^(10k-3))(1-x^(10k-4))(1-x^(10k-6))(1-x^(10k-7))).
Given g.f. k=A(x) then k((1-k)/(1+k))^2 = B(x), k^2((1+k)/(1-k)) = B( x^2) where B(x) = g.f. A078905.
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PROGRAM
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(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( prod(k=1, n, (1-x^k+A)^[0, 1, 1, -1, -1, 0, -1, -1, 1, 1][k%10+1]), n))}
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CROSSREFS
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Sequence in context: A117963 A112803 A124242 this_sequence A082054 A044943 A102395
Adjacent sequences: A112271 A112272 A112273 this_sequence A112275 A112276 A112277
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Aug 30 2005
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