A260631 Denominators of first derivatives of Catalan numbers (as continuous functions of n).

1, 2, 3, 12, 30, 30, 105, 840, 252, 1260, 6930, 1980, 12870, 2574, 2145, 34320, 291720, 79560, 151164, 1511640, 406980, 4476780, 51482970, 13728792, 171609900, 318704100, 84362850, 1181079900, 311375610, 81940950, 1270084725, 40642711200, 10644519600

Offset: 0

Author

Vladimir Reshetnikov, Nov 11 2015

Keywords

• nonn
• frac

Comments

Let C(n) = 4^n*Gamma(n+1/2)/(sqrt(Pi)*Gamma(n+2)), then C'(n) = C(n)*(H(n-1/2) - H(n+1) + log(4)), where H(n) = Sum_{k>=1} (1/k-1/(n+k)) are harmonic numbers.

Examples

			For n = 3, C'(3) = 59/12, so a(3) = denominator(59/12) = 12.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A260630 (numerators), A000108.

Programs

Mathematica

Denominator@FunctionExpand@Table[CatalanNumber'[n] , {n, 0, 32}]

Formula

a(n) = denominator(d(n)), where d(n) satisfies recurrence: d(0) = -1, d(1) = 1/2, (n+1)^2*d(n) = 2*(4*n^2-2*n-1)*d(n-1) - 4*(2*n-3)^2*d(n-2).