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

0,2

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.

Links

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

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).

Example

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

Mathematica

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

Crossrefs

Cf. A260630 (numerators), A000108.

Sequence in context: A359221 A228501 A089414 * A374333 A195913 A048085

Adjacent sequences: A260628 A260629 A260630 * A260632 A260633 A260634

Keyword

nonn,frac

Author

Vladimir Reshetnikov, Nov 11 2015

Status

approved