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A002195
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Numerators of coefficients for numerical integration.
(Formerly M4809 N2056)
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8
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1, -1, 11, -191, 2497, -14797, 92427157, -36740617, 61430943169, -23133945892303, 16399688681447, -3098811853954483, 312017413700271173731, -69213549869569446541, 53903636903066465730877, -522273861988577772410712439, 644962185719868974672135609261
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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]
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REFERENCES
| 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.
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.
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FORMULA
| 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]
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EXAMPLE
| 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]
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MAPLE
| 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]
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CROSSREFS
| 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: A001408 A185123 A036936 * A171553 A068649 A158509
Adjacent sequences: A002192 A002193 A002194 * A002196 A002197 A002198
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KEYWORD
| sign,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2005
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