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A260631 Denominators of first derivatives of Catalan numbers (as continuous functions of n). 2
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 (list; graph; refs; listen; history; text; internal format)
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: A325628 A228501 A089414 * A195913 A048085 A069062

Adjacent sequences:  A260628 A260629 A260630 * A260632 A260633 A260634

KEYWORD

nonn,frac

AUTHOR

Vladimir Reshetnikov, Nov 11 2015

STATUS

approved

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Last modified November 19 03:40 EST 2019. Contains 329310 sequences. (Running on oeis4.)