0,2

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]

H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

T. D. Noe, Table of n, a(n) for n=0..100

Numerators of coefficients in expansion of 1/x-1/sqrt(x)/arcsin(sqrt(x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 11 2002

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]

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]

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]

Cf. A002198.

See A000367, A006954, A008956 and A002671 for underlying sequences. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)

Factor of the LS1[ -2,n] matrix coefficients in A160487.

(End)

Sequence in context: A081421 A121824 A120287 this_sequence A070148 A097499 A132541

Adjacent sequences: A002194 A002195 A002196 this_sequence A002198 A002199 A002200

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 11 2002

Maple program aligned with offset by Johannes W. Meijer (meijgia(AT)hotmail.com), May 15 2009