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A180048
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Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.
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7
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1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 8, 0, 9, 0, 1, 0, 33, 0, 14, 0, 1, 48, 0, 87, 0, 20, 0, 1, 0, 279, 0, 185, 0, 27, 0, 1, 384, 0, 975, 0, 345, 0, 35, 0, 1, 0, 2895, 0, 2640, 0, 588, 0, 44, 0, 1, 3840, 0, 12645, 0, 6090, 0, 938, 0, 54, 0, 1, 0, 35685, 0, 41685, 0, 12558, 0, 1422, 0, 65
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OFFSET
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0,4
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COMMENTS
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Equivalence to the recurrence formula needs formal proof. This continued fraction converges to 0.525135276160981... for w=1. A conjecture by Ramanujan puts this equal to -1 + 1/(sqrt(e*Pi/2) - Sum_{k>=1} 1/(2k-1)!!).
Let us denote the continued fraction by U(w).
Then it is easy to show that Mill's ratio, R(w) = (1 - Phi(w))/f(w), where Phi is the standard normal distribution function and f is the standard normal density function, satisfies R(w) = 1/(w + U(w)).
Indeed, R(w) = 1/(w+1/(w+2/(w+3/(w+... Then we find U(w) = 1/R(w) - w. It was proved in Alexander Kreinin (arXiv:1405.5852) that R(w+t) + Q(w, t) = exp(w*t + w^2/2)*R(t), where Q(w,t) = Sum_{k>=0} Sum_{m=0..k} q(k,m) * t^m * w^(k+1)/(k+1)!.
Substituting t=0, we obtain R(w) = exp(w^2/2)*sqrt(Pi/2) - Sum_{n>=0} w^(2n+1)/(2n+1)!!. If w=1 we obtain Ramanujan's formula. (End)
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LINKS
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FORMULA
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p(0)=1; p(1)=w; p(n) = w*p(n-1) + n*p(n-2) (conjecture).
T(n,k) = T(n-1,k-1) + n*T(n-2,k), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1. - Philippe Deléham, Oct 28 2013
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EXAMPLE
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The denominator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 48 + 87w^2 + 20w^4 + w^6.
Triangle begins
1;
0, 1;
2, 0, 1;
0, 5, 0, 1;
8, 0, 9, 0, 1;
0, 33, 0, 14, 0, 1;
48, 0, 87, 0, 20, 0, 1;
0, 279, 0, 185, 0, 27, 0, 1;
384, 0, 975, 0, 345, 0, 35, 0, 1;
0, 2895, 0, 2640, 0, 588, 0, 44, 0, 1;
3840, 0, 12645, 0, 6090, 0, 938, 0, 54, 0, 1;
0, 35685, 0, 41685, 0, ... (End)
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MATHEMATICA
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Table[ CoefficientList[ Denominator[ Together[ Fold[ #2/(w+#1) &, Infinity, Reverse @ Table[ k, {k, 1, n} ] ] ] ], w ], {n, 16} ] (* or equivalently *) Clear[ p ]; p[ 0 ]=1; p[ 1 ]=w; p[ n_ ]:=p[ n ]= w*p[ n-1 ] + n*p[ n-2 ]; Table[ CoefficientList[ p[ k ]//Expand, w ], {k, 0, 15} ]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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