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

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, 40320, 15120, 1200, 20, 0, 362880, 141120, 12600, 300, 1, 0, 3628800, 1451520, 141120, 4200, 30, 0, 39916800, 16329600, 1693440, 58800, 630, 1, 0, 479001600

Offset: 0

Author

Wouter Meeussen, Aug 08 2010

Keywords

• nonn
• tabf

Comments

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.

Examples

			Triangle starts:
  0;
  0,   1;
  0,   2;
  0,   6,   1;
  0,  24,   6;
  0, 120,  36,  1;
  0, 720, 240, 12;
The numerator of w/(1+w/(2+w/(3+w/(4+w/5)))) equals 120*w + 36*w^2 + w^3.
		

Crossrefs

Variant: A221913.

Cf. A084950, A180048, A180049, A008297, A111596, A105278, A052119.

Programs

Mathematica

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]}]

Formula

T(n,m) = (n-m+1)!/m!*binomial(n-m, m-1) for n >= 0, 0 <= m <= (n+1)/2.