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

0,5

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.

Links

Table of n, a(n) for n=0..49.

Formula

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

Example

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.

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]}]

Crossrefs

Variant: A221913.

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

Sequence in context: A137437 A183189 A330609 * A180397 A347133 A317842

Adjacent sequences: A180044 A180045 A180046 * A180048 A180049 A180050

Keyword

nonn,tabf

Author

Wouter Meeussen, Aug 08 2010

Status

approved