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

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, 561, 0, 285, 0, 33, 0, 1, 945, 0, 1830, 0, 510, 0, 42, 0, 1, 0, 6555, 0, 4680, 0, 840, 0, 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

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..78.

Hungarian discussion forum

Formula

b(0)=1; b(1)=w; b(n) = w*b(n-1) + (n+1)*b(n-2) (conjecture).

Example

The numerator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 57w + 18w^3 + w^5.
From Philippe Deléham, Nov 06 2013: (Start)
Triangle begins:
      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,  561,     0,  285,     0,  33,    0,  1;
    945,    0,  1830,    0,   510,   0,   42,  0,  1;
      0, 6555,     0, 4680,     0, 840,    0, 52,  0, 1;
  10395,    0, 26685,    0, 10290,   0, 1302,  0, 63, 0, 1;
  ... (End)
[extended by M. F. Hasler, Oct 21 2014]

Mathematica

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} ]

Prog

(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

Crossrefs

Cf. A059480, A084950, A180047, A180048, A230698.

Sequence in context: A363978 A262964 A135481 * A244454 A238123 A128311

Adjacent sequences: A180046 A180047 A180048 * A180050 A180051 A180052

Keyword

nonn,tabl

Author

Wouter Meeussen, Aug 08 2010

Status

approved