Search: id:A002195
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%I A002195 M4809 N2056
%S A002195 1,1,11,191,2497,14797,92427157,36740617,61430943169,23133945892303,
%T A002195 16399688681447,3098811853954483,312017413700271173731,69213549869569446541,
%U A002195 53903636903066465730877,522273861988577772410712439,644962185719868974672135609261
%V A002195 1,-1,11,-191,2497,-14797,92427157,-36740617,61430943169,-23133945892303,
%W A002195 16399688681447,-3098811853954483,312017413700271173731,-69213549869569446541,
%X A002195 53903636903066465730877,-522273861988577772410712439,644962185719868974672135609261
%N A002195 Numerators of coefficients for numerical integration.
%C A002195 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]
%D A002195 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002195 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002195 H. E. Salzer, Coefficients for numerical integration with central differences, 
               Phil. Mag., 35 (1944), 262-264.
%D A002195 H. E. Salzer, Coefficients for repeated integration with central differences, 
               Journal of Mathematics and Physics, 28 (1949), 54-61.
%D A002195 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, 
               Vol. 2, Engelmann, Leipzig, 1880, p. 545.
%F A002195 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
%F A002195 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]
%e A002195 a(1)=-1 because (1/3)*int(t*(t^2-1^2),t=0..1)=-1/12.
%e A002195 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]
%p A002195 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)
%p A002195 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]
%Y A002195 Cf. A002196.
%Y A002195 See A000367, A006954, A008955 and A009445 for underlying sequences. [From 
               Johannes W. Meijer (meijgia(AT)hotmail.com), Jan 27 2009]
%Y A002195 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%Y A002195 Factor of ZS1[ -1,n] matrix coefficients in A160474.
%Y A002195 (End)
%Y A002195 Sequence in context: A034787 A001408 A036936 this_sequence A068649 A158509 
               A072290
%Y A002195 Adjacent sequences: A002192 A002193 A002194 this_sequence A002196 A002197 
               A002198
%K A002195 sign,frac
%O A002195 0,3
%A A002195 N. J. A. Sloane (njas(AT)research.att.com).
%E A002195 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2005

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