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A180049 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... .

%I A180049 #17 Dec 24 2021 21:54:25
%S A180049 1,0,1,3,0,1,0,7,0,1,15,0,12,0,1,0,57,0,18,0,1,105,0,141,0,25,0,1,0,
%T A180049 561,0,285,0,33,0,1,945,0,1830,0,510,0,42,0,1,0,6555,0,4680,0,840,0,
%U A180049 52,0,1,10395,0,26685,0,10290,0,1302,0,63,0,1,0,89055,0,82845,0,20370,0,1926,0,75,0,1
%N A180049 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... .
%C A180049 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)!!). Row sums equal A059480.
%H A180049 <a href="http://www.komal.hu/forum/forum.cgi?a=to&amp;tid=94&amp;st=50&amp;dr=1&amp;sp=1233">Hungarian discussion forum</a>
%F A180049 b(0)=1; b(1)=w; b(n) = w*b(n-1) + (n+1)*b(n-2) (conjecture).
%e A180049 The numerator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 57w + 18w^3 + w^5.
%e A180049 From _Philippe Deléham_, Nov 06 2013: (Start)
%e A180049 Triangle begins:
%e A180049       1;
%e A180049       0,    1;
%e A180049       3,    0,     1;
%e A180049       0,    7,     0,    1;
%e A180049      15,    0,    12,    0,     1;
%e A180049       0,   57,     0,   18,     0,   1;
%e A180049     105,    0,   141,    0,    25,   0,    1;
%e A180049       0,  561,     0,  285,     0,  33,    0,  1;
%e A180049     945,    0,  1830,    0,   510,   0,   42,  0,  1;
%e A180049       0, 6555,     0, 4680,     0, 840,    0, 52,  0, 1;
%e A180049   10395,    0, 26685,    0, 10290,   0, 1302,  0, 63, 0, 1;
%e A180049   ... (End)
%e A180049 [extended by _M. F. Hasler_, Oct 21 2014]
%t A180049 Table[ CoefficientList[ Numerator[ Together[ Fold[ #2/(w+#1) &, Infinity, Reverse @ Table[ k, {k, 1, n} ] ] ] ], w ], {n, 2, 16} ] or equivalently Clear[ b ]; b[ 0 ]=1; b[ 1 ]=w; b[ n_ ]:=b[ n ] = w b[ n-1 ]+(n+1) b[ n-2 ]; Table[ CoefficientList[ b[ k ]//Expand, w ], {k, 0, 14} ]
%o A180049 (PARI) t=x-w;for(n=1,12,t=substpol(t,x,w+n/x);print(Vecrev(numerator(substpol(t,x,w))))) \\ _M. F. Hasler_, Oct 21 2014
%Y A180049 Cf. A059480, A084950, A180047, A180048, A230698.
%K A180049 nonn,tabl
%O A180049 1,4
%A A180049 _Wouter Meeussen_, Aug 08 2010