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%I #15 Apr 09 2017 03:40:03
%S 1,2,3,12,30,30,105,840,252,1260,6930,1980,12870,2574,2145,34320,
%T 291720,79560,151164,1511640,406980,4476780,51482970,13728792,
%U 171609900,318704100,84362850,1181079900,311375610,81940950,1270084725,40642711200,10644519600
%N Denominators of first derivatives of Catalan numbers (as continuous functions of n).
%C 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.
%H G. C. Greubel, <a href="/A260631/b260631.txt">Table of n, a(n) for n = 0..1000</a>
%F 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).
%e For n = 3, C'(3) = 59/12, so a(3) = denominator(59/12) = 12.
%t Denominator@FunctionExpand@Table[CatalanNumber'[n] , {n, 0, 32}]
%Y Cf. A260630 (numerators), A000108.
%K nonn,frac
%O 0,2
%A _Vladimir Reshetnikov_, Nov 11 2015
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