cp's OEIS Frontend

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

%I A220422 #11 Mar 16 2020 14:28:18
%S A220422 5,-1,65,-1381,50525,-2702761,199360985,-19391512141,2404879675445,
%T A220422 -370371188237521,69348874393137905,-15514534163557086901,
%U A220422 4087072509293123892365,-1252259641403629865468281,441543893249023104553682825,-177519391579539289436664789661
%N A220422 Numerators of coefficients of an expansion of the logarithm of the Catalan numbers.
%C A220422 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)).
%D A220422 Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.
%H A220422 J. L. Fields, <a href="https://doi.org/10.1017/S0013091500013171">A note on the asymptotic expansion of a ratio of gamma functions</a>, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
%H A220422 D. Kessler and J. Schiff, <a href="http://u.math.biu.ac.il/~schiff/Papers/prepap3.pdf">The asymptotics of factorials, binomial coefficients and Catalan numbers</a>. April 2006.
%F A220422 a(n) = -4^(2*n+1)*B_{2*n+1}(-1/4)/(2*n+1), B_{n}(x) the Bernoulli polynomials.
%F A220422 a(n) = 4 - E(2*n), E the Euler numbers A122045.
%e A220422 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).
%p A220422 A220422 := n -> 4 - euler(2*n):
%o A220422 (Sage)
%o A220422 def A220422Generator() :
%o A220422     A = {-1:0, 0:1};
%o A220422     k = 0; e = 1; i = 0; m = 0
%o A220422     while True:
%o A220422         An = 0; A[k + e] = 0; e = -e
%o A220422         for j in (0..i) :
%o A220422             An += A[k]; A[k] = An; k += e
%o A220422         if e < 0 :
%o A220422             yield 4 - A[-m]*(-1)^m
%o A220422             m += 1
%o A220422         i += 1
%o A220422 A220422 = A220422Generator()
%o A220422 [next(A220422) for n in (1..16)]
%Y A220422 Cf. A000108, A122045.
%Y A220422 The exponential version is A220002.
%K A220422 sign
%O A220422 1,1
%A A220422 _Peter Luschny_, Dec 28 2012