0,3

The numerators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{_2k}, the central factorial numbers t(2n,2n-2k) and the factor (2n+1)!. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]

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

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

H. E. Salzer, Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264.

H. E. Salzer, Coefficients for repeated integration with central differences, Journal of Mathematics and Physics, 28 (1949), 54-61.

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

a(n)=numerator of (2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..s), t=0..1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2005

a(0) = 1; a(n) = numerator of sum((-1)^(k+n+1)*(B{_2k}/(2k))*t(2n,2n-2k+2), k=1..n)/(2n-1)! for n=1,2,3,... [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]

a(1)=-1 because (1/3)*int(t*(t^2-1^2),t=0..1)=-1/12.

a(3) = numer((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = -191; [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]

a:=n->numer((2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1)): seq(a(n), n=0..16); (Deutsch)

nmax:=10: jn:=nmax: im:=nmax: Omega[0]:=1: for n from 1 to nmax do for j from 1 to jn do cfn1[1, j]:=1 end do: for i from 2 to im do cfn1[i, 1]:=0 end do: for j from 2 to jn do for i from 2 to im do cfn1[i, j]:=cfn1[i-1, j-1]*(j-1)^2+cfn1[i, j-1] end do end do: Omega[n]:= (sum((-1)^(k+n+1)*(bernoulli(2*k)/(2*k))*cfn1[n-k+1, n], k=1..n))/(2*n-1)! end do: a:=n-> numer(Omega[n]): seq(a(n), n=0..nmax); [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]

Cf. A002196.

See A000367, A006954, A008955 and A009445 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 ZS1[ -1,n] matrix coefficients in A160474.

(End)

Sequence in context: A034787 A001408 A036936 this_sequence A068649 A158509 A072290

Adjacent sequences: A002192 A002193 A002194 this_sequence A002196 A002197 A002198

sign,frac

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

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2005