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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/... 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 A317842 A021489 Adjacent sequences:  A180044 A180045 A180046 * A180048 A180049 A180050 KEYWORD nonn,tabf AUTHOR Wouter Meeussen, Aug 08 2010 STATUS approved

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Last modified January 19 13:17 EST 2021. Contains 340270 sequences. (Running on oeis4.)