Search: id:A002196
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%I A002196 M4880 N2093
%S A002196 1,12,720,60480,3628800,95800320,2615348736000,4483454976000,
%T A002196 32011868528640000,51090942171709440000,152579284313702400000,
%U A002196 120866571766215475200000,50814724101952310083584000000
%N A002196 Denominators of coefficients for numerical integration.
%C A002196 The denominators 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]
%D A002196 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002196 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002196 H. E. Salzer, Coefficients for numerical integration with central differences, 
               Phil. Mag., 35 (1944), 262-264.
%D A002196 H. E. Salzer, Coefficients for repeated integration with central differences, 
               Journal of Mathematics and Physics, 28 (1949), 54-61.
%D A002196 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, 
               Vol. 2, Engelmann, Leipzig, 1880, p. 545.
%F A002196 a(n)=denominator of (2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1): 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2005
%F A002196 a(0) = 1; a(n) = denominator 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]
%e A002196 a(1)=12 because (1/3)*int(t*(t^2-1^2),t=0..1)=-1/12.
%e A002196 a(3) = denom((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so 
               a(3) = 60480; [From Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Jan 27 2009]
%p A002196 a:=n->denom((2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1)): seq(a(n),
               n=0..14); (Deutsch)
%p A002196 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-> denom(Omega[n]): 
               seq(a(n),n=0..nmax); [From Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Jan 27 2009]
%Y A002196 Cf. A002195.
%Y A002196 See A000367, A006954, A008955 and A009445 for underlying sequences. [From 
               Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%Y A002196 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%Y A002196 Factor of ZS1[ -1,n] matrix coefficients in A160474.
%Y A002196 (End)
%Y A002196 Sequence in context: A126159 A071307 A060612 this_sequence A141421 A000909 
               A162447
%Y A002196 Adjacent sequences: A002193 A002194 A002195 this_sequence A002197 A002198 
               A002199
%K A002196 nonn,frac
%O A002196 0,2
%A A002196 N. J. A. Sloane (njas(AT)research.att.com).
%E A002196 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2005

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