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A220422
Numerators of coefficients of an expansion of the logarithm of the Catalan numbers.
3
5, -1, 65, -1381, 50525, -2702761, 199360985, -19391512141, 2404879675445, -370371188237521, 69348874393137905, -15514534163557086901, 4087072509293123892365, -1252259641403629865468281, 441543893249023104553682825, -177519391579539289436664789661
OFFSET
1,1
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)).
REFERENCES
Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.
LINKS
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.
EXAMPLE
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).
MAPLE
A220422 := n -> 4 - euler(2*n):
PROG
(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)]
CROSSREFS
The exponential version is A220002.
Sequence in context: A050970 A335955 A138548 * A251596 A294258 A294260
KEYWORD
sign
AUTHOR
Peter Luschny, Dec 28 2012
STATUS
approved