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A002195 Numerators of coefficients for numerical integration.

%I A002195 M4809 N2056 #59 Jun 01 2024 11:44:34
%S A002195 1,-1,11,-191,2497,-14797,92427157,-36740617,61430943169,
%T A002195 -23133945892303,16399688681447,-3098811853954483,
%U A002195 312017413700271173731,-69213549869569446541,53903636903066465730877,-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 A008955(n, k) and the factor (2n+1)!. - _Johannes W. Meijer_, Jan 27 2009
%C A002195 These numbers are the numerators of the constant term in the Laurent expansion of the cosech^(2n)(x)/2^(2n) function. - _Istvan Mezo_, Apr 21 2023
%D A002195 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002195 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002195 Iaroslav V. Blagouchine, <a href="http://math.colgate.edu/~integers/sjs3/sjs3.Abstract.html">Three notes on Ser's and Hasse's representation for the zeta-functions</a>, Integers (2018) 18A, Article #A3.
%H A002195 H. E. Salzer, <a href="/A002195/a002195.pdf">Coefficients for numerical integration with central differences</a>, Phil. Mag., 35 (1944), 262-264. [Annotated scanned copy]
%H A002195 H. E. Salzer, <a href="https://doi.org/10.1080/14786444408520877">XXXII. Coefficients for numerical integration with central differences</a>, Phil. Mag., 35 (1944), 262-264.
%H A002195 H. E. Salzer, <a href="https://doi.org/10.1002/sapm194928154">Coefficients for repeated integration with central differences</a>, Journal of Mathematics and Physics, 28 (1949), 54-61.
%H A002195 T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/545/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 545.
%F A002195 a(n) = numerator of (2/(2*n+1)!)*Integral_{t=0..1} t*Product_{k=1..n} t^2-k^2. - _Emeric Deutsch_, Jan 25 2005
%F A002195 a(0) = 1; a(n) = numerator [sum((-1)^(k+n+1) * (B{2k}/(2*k)) * A008955(n-1, n-k), k = 1..n)/(2*n-1)!] for n >= 1. - _Johannes W. Meijer_, Jan 27 2009
%F A002195 a(n) = numerator(sum(k=1..2*n, binomial(2*n+k-1,2*n-1)*sum(j=1..k, (binomial(k,j)*sum(i=0,j/2, (2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(-i)))/(2^j*(2*n+j)!)))), n>0, a(0)=1. - _Vladimir Kruchinin_, Feb 04 2013
%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. - _Johannes W. Meijer_, 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); # _Emeric Deutsch_, Feb 20 2005
%p A002195 nmax:=16: with(combinat): A008955 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j = -k..k) end proc: Omega(0) := 1: for n from 1 to nmax do Omega(n) := sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1))* A008955(n-1,n-k1), k1=1..n)/(2*n-1)! end do: a := n-> numer(Omega(n)): seq(a(n), n=0..nmax); # _Johannes W. Meijer_, Jan 27 2009, Revised Sep 21 2012
%t A002195 a[0] = 1; a[n_] := Sum[Binomial[2*n+k-1, 2*n-1]*Sum[Binomial[k, j]*Sum[(2*i-j)^(2*n+j)*Binomial[j, i]*(-1)^(-i), {i, 0, j/2}]/(2^j*(2*n+j)!), {j, 1, k}], {k, 1, 2*n}]; Table[a[n] // Numerator, {n, 0, 16}] (* _Jean-François Alcover_, Apr 18 2014, after _Vladimir Kruchinin_ *)
%t A002195 a[n_] := Numerator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* _Istvan Mezo_, Apr 21 2023 *)
%o A002195 (Maxima)
%o A002195 a(n):=num(sum(binomial(2*n+k-1,2*n-1)*sum((binomial(k,j)*sum((2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(-i),i,0,j/2))/(2^j*(2*n+j)!),j,1,k),k,1,2*n)); /* _Vladimir Kruchinin_, Feb 04 2013 */
%Y A002195 Cf. A002196.
%Y A002195 See A000367, A006954, A008955 and A009445 for underlying sequences.
%K A002195 sign,frac
%O A002195 0,3
%A A002195 _N. J. A. Sloane_
%E A002195 More terms from _Emeric Deutsch_, Jan 25 2005
%E A002195 Edited by _Johannes W. Meijer_, Sep 21 2012