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%I #12 Feb 11 2020 02:05:33
%S 0,0,1,0,2,0,6,1,0,24,6,0,120,36,1,0,720,240,12,0,5040,1800,120,1,0,
%T 40320,15120,1200,20,0,362880,141120,12600,300,1,0,3628800,1451520,
%U 141120,4200,30,0,39916800,16329600,1693440,58800,630,1,0,479001600
%N Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/3 + w/...
%C Equivalence to the binomial formula needs formal proof. This c.f. converges to A052119 = 0.697774657964.. = BesselI(1,2)/BesselI(0,2) for w = 1.
%F T(n,m) = (n-m+1)!/m!*binomial(n-m, m-1) for n >= 0, 0 <= m <= (n+1)/2.
%e Triangle starts:
%e 0;
%e 0, 1;
%e 0, 2;
%e 0, 6, 1;
%e 0, 24, 6;
%e 0, 120, 36, 1;
%e 0, 720, 240, 12;
%e .
%e The numerator of w/(1+w/(2+w/(3+w/(4+w/5)))) equals 120*w + 36*w^2 + w^3.
%t Table[CoefficientList[Numerator[Together[Fold[w/(#2+#1) &,Infinity,Reverse @ Table[k,{k,1,n}]]]],w],{n,16}]; (* or equivalently *) Table[(n-m+1)!/m! *Binomial[n-m,m-1], {n,0,16}, {m,0,Floor[n/2+1/2]}]
%Y Variant: A221913.
%Y Cf. A084950, A180048, A180049, A008297, A111596, A105278, A052119.
%K nonn,tabf
%O 0,5
%A _Wouter Meeussen_, Aug 08 2010
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