A220422 Numerators of coefficients of an expansion of the logarithm of the Catalan numbers.

5, -1, 65, -1381, 50525, -2702761, 199360985, -19391512141, 2404879675445, -370371188237521, 69348874393137905, -15514534163557086901, 4087072509293123892365, -1252259641403629865468281, 441543893249023104553682825, -177519391579539289436664789661

Offset: 1

Author

Peter Luschny, Dec 28 2012

Keywords

• sign

Comments

Let C(n) denote the Catalan numbers A000108 and S(n) = Sum_{k>=1} a(k)/(2*k*(4*n+3)^(2*k)) then log(C(n)) = (1/2)*(n*log(16)-3*log(n+3/4)-log(Pi)+S(n)).

Examples

			Let N = 4*n+3 then log(C(n)) = (n*log(16)-3*log(n+3/4)-log(Pi))/2+a(1)/(4*N^2)+a(2)/(8*N^4)+a(3)/(12*N^6)+a(4)/(16*N^8)+O(1/N^10).
		

References

• Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.

Links

• J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
• D. Kessler and J. Schiff, The asymptotics of factorials, binomial coefficients and Catalan numbers. April 2006.

Crossrefs

Cf. A000108, A122045.

The exponential version is A220002.

Programs

Maple

A220422 := n -> 4 - euler(2*n):

Sage

def A220422Generator() :
    A = {-1:0, 0:1};
    k = 0; e = 1; i = 0; m = 0
    while True:
        An = 0; A[k + e] = 0; e = -e
        for j in (0..i) :
            An += A[k]; A[k] = An; k += e
        if e < 0 :
            yield 4 - A[-m]*(-1)^m
            m += 1
        i += 1
A220422 = A220422Generator()
[next(A220422) for n in (1..16)]

Formula

a(n) = -4^(2*n+1)*B_{2*n+1}(-1/4)/(2*n+1), B_{n}(x) the Bernoulli polynomials.

a(n) = 4 - E(2*n), E the Euler numbers A122045.