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Search: id:A158691
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| A158691 |
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Expansion of 1 + (1-x) + (1-x)(1-x^3) + (1-x)(1-x^3)(1-x^5) + ... about x = 1. |
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+0
2
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| 1, 1, 3, 12, 61, 380, 2815, 24213, 237348, 2612681, 31915787, 428481472, 6271362282, 99388642292, 1695614865711, 30984649882928, 603790447393402, 12498732438500663, 273902239550757626, 6334968666307580051
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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There appears to be a connection with the alternating series 1 - (1-x) + (1-x)(1-x^2) - (1-x)(1-x^2)(1-x^3) + .... If we replace x with 1/(1-x) in the partial sum 1 - (1-x) + (1-x)(1-x^2) - (1-x)(1-x^2)(1-x^3) + ... + (-1)^n(1-x)(1-x^2)...(1-x^n) and then expand about x = 0 we seem to get a series whose first n+1 coefficients agree with the first n+1 terms of the present sequence.
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FORMULA
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Sum {n = 0..inf} Product {i= 1..n} (1-(1-x)^(2*i-1)) = 1 + x + 3*x^2 + 12*x^3 + 61*x^4 + .... Compare with A022493, A138265 and A158690.
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MAPLE
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1) g:=sum(product(1-(1-x)^(2*i-1), i= 1..n), n = 0..20): gser:=series(g, x = 0, 20): seq(coeff(gser, x, n), n = 0..19);
2) Conjecturally
g:=sum((-1)^n*product(1-1/(1-x)^i, i= 1..n), n = 0..20): gser:=series(g, x = 0, 20): seq(coeff(gser, x, n), n = 0..19);
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CROSSREFS
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A022493, A138265, A158690.
Sequence in context: A161799 A159925 A121694 this_sequence A038171 A074516 A045740
Adjacent sequences: A158688 A158689 A158690 this_sequence A158692 A158693 A158694
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KEYWORD
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easy,nonn
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AUTHOR
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Peter Bala (pbala(AT)talktalk.net), Mar 24 2009
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