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A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

%I A220002 #50 Mar 16 2020 14:26:42
%S A220002 1,5,21,715,-162877,19840275,-7176079695,1829885835675,
%T A220002 -5009184735027165,2216222559226679575,-2463196751104762933637,
%U A220002 1679951011110471133453965,-5519118103058048675551057049,5373485053345792589762994345215,-12239617587594386225052760043303511
%N A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.
%C A220002 Let N = 4*n+3 and A = sum_{k>=0} a(k)/(A123854(k)*N^(2*k)) then
%C A220002 C(n) ~ 8*4^n*A/(N*sqrt(N*Pi)), C(n) = (4^n/sqrt(Pi))*(Gamma(n+1/2)/ Gamma(n+2)) the Catalan numbers A000108.
%C A220002 The asymptotic expansion of the Catalan numbers considered here is based on the Taylor expansion of square root of the sine cardinal. This asymptotic series involves only even powers of N, making it more efficient than the asymptotic series based on Stirling's approximation to the central binomial which involves all powers (see for example: D. E. Knuth, 7.2.1.6 formula (16)). The series is discussed by Kessler and Schiff but is included as a special case in the asymptotic expansion given by J. L. Fields for quotients Gamma(x+a)/Gamma(x+b) and discussed by Y. L. Luke (p. 34-35), apparently overlooked by Kessler and Schiff.
%D A220002 Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees—History of Combinatorial Generation, 2006.
%D A220002 Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.
%H A220002 Peter Luschny, <a href="/A220002/b220002.txt">Table of n, a(n) for n = 0..99</a>
%H A220002 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 A220002 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 A220002 Let [x^n]T(f(x)) denote the coefficient of x^n in the Taylor expansion of f(x) then r(n) = (-1)^n*prod_{i=1..2n}(2i+1)*[x^(2*n)]T(sqrt(sin(x)/x)) is the rational coefficient of the asymptotic expansion (in N=4*n+3) and a(n) = numerator(r(n)) = r(n)*2^(3*n-bs(n)), where bs(n) is the binary sum of n (A000120).
%F A220002 Also a(n) = numerator([x^(2*n)]T(exp(S))) where S = sum_{k>=1}((4-E(2*k))/ (4*k)*x^(2*k)) and E(n) the Euler numbers A122045.
%F A220002 Also a(n) = sf(4*n+1)*2^(3*n-bs(n))*F_{2*n}(-1/4) where sf(n) is the swinging factorial A056040, bs(n) the binary sum of n and F_{n}(x) J. L. Fields' generalized Bernoulli polynomials A220412.
%F A220002 In terms of sequences this means
%F A220002 r(n) = (-1)^n*A103639(n)*A008991(n)/A008992(n),
%F A220002 a(n) = (-1)^n*A220371(n)*A008991(n)/A008992(n).
%F A220002 Note that a(n) = r(n)*A123854(n) and A123854(n) = 2^A004134(n) = 8^n/2^A000120(n).
%F A220002 Formula from _Johannes W. Meijer_:
%F A220002 a(n) = d(n+1)*A098597(2*n+1)*(A008991(n)/A008992(n)) with d(1) = 1 and
%F A220002 d(n+1) = -4*(2*n+1)*A161151(n)*d(n),
%F A220002 d(n+1) = (-1)^n*2^(-1)*(2*(n+1))!*A060818(n)*A048896(n).
%e A220002 With N = 4*n+3 the first few terms of A are A = 1 + 5/(4*N^2) + 21/(32*N^4) + 715/(128*N^6) - 162877/(2048*N^8) + 19840275/(8192*N^10). With this A C(n) = round(8*4^n*A/(N*sqrt(N*Pi))) for n = 0..39 (if computed with sufficient numerical precision).
%p A220002 A220002 := proc(n) local s; s := n -> `if`(n > 0, s(iquo(n,2))+n, 0);
%p A220002 (-1)^n*mul(4*i+2, i = 1..2*n)*2^s(iquo(n,2))*coeff(taylor(sqrt(sin(x)/x), x,2*n+2), x, 2*n) end: seq(A220002(n), n = 0..14);
%p A220002 # Second program illustrating J. L. Fields expansion of gamma quotients.
%p A220002 A220002 := proc(n) local recF, binSum, swing;
%p A220002 binSum := n -> add(i,i=convert(n,base,2));
%p A220002 swing := n -> n!/iquo(n, 2)!^2;
%p A220002 recF := proc(n, x) option remember; `if`(n=0, 1, -2*x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(2*k+2)*recF(n-2*k-2,x),k=0..n/2-1)) end: recF(2*n,-1/4)*2^(3*n-binSum(n))*swing(4*n+1) end:
%t A220002 max = 14; CoefficientList[ Series[ Sqrt[ Sinc[x]], {x, 0, 2*max+1}], x^2][[1 ;; max+1]]*Table[ (-1)^n*Product[ (2*k+1), {k, 1, 2*n}], {n, 0, max}] // Numerator (* _Jean-François Alcover_, Jun 26 2013 *)
%o A220002 (Sage)
%o A220002 length = 15; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
%o A220002 def A005187(n): return A005187(n//2) + n if n > 0 else 0
%o A220002 def A220002(n):
%o A220002     P = mul(4*i+2 for i in (1..2*n)) << A005187(n//2)
%o A220002     return (-1)^n*P*T.coefficient(x, 2*n)
%o A220002 [A220002(n) for n in range(length)]
%o A220002 (Sage) # Second program illustrating the connection with the Euler numbers.
%o A220002 def A220002_list(n):
%o A220002     S = lambda n: sum((4-euler_number(2*k))/(4*k*x^(2*k)) for k in (1..n))
%o A220002     T = taylor(exp(S(2*n+1)),x,infinity,2*n-1).coefficients()
%o A220002     return [t[0].numerator() for t in T][::-1]
%o A220002 A220002_list(15)
%Y A220002 The logarithmic version is A220422. Appears in A193365 and A220466.
%Y A220002 Cf. A220412.
%K A220002 sign
%O A220002 0,2
%A A220002 _Peter Luschny_, Dec 27 2012