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A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).

%I A180047 #16 May 11 2025 02:36:27
%S A180047 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 A180047 40320,15120,1200,20,0,362880,141120,12600,300,1,0,3628800,1451520,
%U A180047 141120,4200,30,0,39916800,16329600,1693440,58800,630,1,0,479001600
%N A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).
%C A180047 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 A180047 T(n,m) = (n-m+1)!/m!*binomial(n-m, m-1) for n >= 0, 0 <= m <= (n+1)/2.
%e A180047 Triangle starts:
%e A180047   0;
%e A180047   0,   1;
%e A180047   0,   2;
%e A180047   0,   6,   1;
%e A180047   0,  24,   6;
%e A180047   0, 120,  36,  1;
%e A180047   0, 720, 240, 12;
%e A180047 The numerator of w/(1+w/(2+w/(3+w/(4+w/5)))) equals 120*w + 36*w^2 + w^3.
%t A180047 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 A180047 Variant: A221913.
%Y A180047 Cf. A084950, A180048, A180049, A008297, A111596, A105278, A052119.
%K A180047 nonn,tabf
%O A180047 0,5
%A A180047 _Wouter Meeussen_, Aug 08 2010