%I A002197 M5049 N2183
%S A002197 1,17,367,27859,1295803,5329242827,25198857127,11959712166949,
%T A002197 11153239773419941,31326450596954510807,3737565567167418110609,
%U A002197 2102602044094540855003573,189861334343507894443216783
%N A002197 Numerators of coefficients for numerical integration.
%C A002197 The numerators of these coefficients for numerical integration are a 
               combination of the Bernoulli numbers B{_2k}, the central factorial 
               numbers 4^(k)t(2n+1,2n+1-2k) and the factor 4^n*(2*n+1)!. [From Johannes 
               W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%D A002197 H. E. Salzer, Coefficients for mid-interval numerical integration with 
               central differences, Phil. Mag., 36 (1945), 216-218.
%D A002197 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002197 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002197 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, 
               Vol. 2, Engelmann, Leipzig, 1880, p. 545.
%H A002197 T. D. Noe, <a href="b002197.txt">Table of n, a(n) for n=0..100</a>
%F A002197 Numerators of coefficients in expansion of 1/x-1/sqrt(x)/arcsin(sqrt(x)). 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 11 2002
%F A002197 a(n) = numerator of sum((1-2^(2*k-1))* (-1)^(k)*(B_{2k}/(2*k))*4^(n-k)*t(2*n-1,
               2*k-1),k=1..n) /(2*4^(n-1)*(2*n-1)!) for n = 0,1,2,3,... [From Johannes 
               W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%e A002197 a(2) = numer(((1-2^1)*(-1)*((1/6)/2)*(9) + (1-2^3)*(1)*((-1/30)/4)*(10) 
               + (1-2^5)*(-1)*((1/42)/6)*(1))/(2*4^2*5!)) so a(2) = 367; [From Johannes 
               W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%p A002197 nmax:=9: jn:=nmax: im:=nmax: for n from 1 to nmax do for i from 2 to 
               im do cfn2[i,1]:=0 end do: for j from 1 to jn do cfn2[1,j]:=1 end 
               do: for j from 2 to jn do for i from 2 to im do cfn2[i,j]:= cfn2[i,
               j-1] + cfn2[i-1,j-1]*(2*j-3)^2 end do end do: Delta[n-1]:=sum((1-2^(2*k-1))* 
               (-1)^(n+1)*(-bernoulli(2*k)/(2*k))*(-1)^(k+n)*cfn2[n-k+1,n],k=1..n) 
               /(2*4^(n-1)*(2*n-1)!) end do: a:=n-> numer(Delta[n]): seq(a(n),n=0..nmax-1); 
               [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%Y A002197 Cf. A002198.
%Y A002197 See A000367, A006954, A008956 and A002671 for underlying sequences. [From 
               Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%Y A002197 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%Y A002197 Factor of the LS1[ -2,n] matrix coefficients in A160487.
%Y A002197 (End)
%Y A002197 Sequence in context: A081421 A121824 A120287 this_sequence A070148 A097499 
               A132541
%Y A002197 Adjacent sequences: A002194 A002195 A002196 this_sequence A002198 A002199 
               A002200
%K A002197 nonn
%O A002197 0,2
%A A002197 N. J. A. Sloane (njas(AT)research.att.com).
%E A002197 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 11 2002
%E A002197 Maple program aligned with offset by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               May 15 2009