A260630 Numerators of first derivatives of Catalan numbers (as continuous functions of n).

-1, 1, 5, 59, 449, 1417, 16127, 429697, 437705, 7549093, 145103527, 146489197, 3396112211, 2442184933, 7369048679, 429556076057, 13374954901367, 13427048535167, 94315062045929, 3500487562166393, 3510273150915593, 144285489968702713, 6218562602767668259

Offset: 0

Author

Vladimir Reshetnikov, Nov 11 2015

Keywords

• sign
• 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) = numerator(59/12) = 59.
		

Links

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

Crossrefs

Cf. A260631 (denominators).

Cf. A000108, A001008, A074599.

Programs

Mathematica

Numerator@FunctionExpand@Table[CatalanNumber'[n] , {n, 0, 22}]

Formula

a(n) = numerator(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).