%I M5049 N2183 #51 Mar 13 2019 11:33:31
%S 1,17,367,27859,1295803,5329242827,25198857127,11959712166949,
%T 11153239773419941,31326450596954510807,3737565567167418110609,
%U 2102602044094540855003573,189861334343507894443216783
%N Numerators of coefficients for numerical integration.
%C The numerators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008956(n, k) and the factor 4^n*(2*n+1)!. - _Johannes W. Meijer_, Jan 27 2009
%D H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002197/b002197.txt">Table of n, a(n) for n=0..100</a>
%H H. E. Salzer, <a href="/A002197/a002197.pdf">Coefficients for mid-interval numerical integration with central differences</a>, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
%H 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 Numerators of coefficients in expansion of 1/x-1/sqrt(x)/arcsin(sqrt(x)). - _Vladeta Jovovic_, Aug 11 2002
%F a(n) = numerator [sum((1-2^(2*k-1)) * (-1)^(k) * (B{2k}/(2*k)) * A008956(n, n+1-k), k=1..n+1) / (2*4^(n)*(2*n+1)!)] for n >= 0. - _Johannes W. Meijer_, Jan 27 2009
%F a(n) = numerator((sum(k=0..2*n-1, binomial(2*n+k-1,2*n-2)*sum(j=1..k+1, (binomial(k+1,j)*sum(i=0..j/2,(2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(n-i)))/(2^(j-1)*(2*n+j)!))))/(2*n-1)). - _Vladimir Kruchinin_, May 16 2013
%e a(2) = numer(((1-2^1)*(-1)*((1/6)/2)*(9) + (1-2^3)*(1)*((-1/30)/4)*(10) + (1-2^5)*(-1)*((1/42)/6)*(1))/(2*4^2*5!)) so a(2) = 367. - _Johannes W. Meijer_, Jan 27 2009
%p nmax:=13: for n from 0 to nmax do A008956(n, 0) := 1: A008956(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do A008956(n, k) := (2*n-1)^2*A008956(n-1, k-1) + A008956(n-1, k) od: od: for n from 0 to nmax do Delta(n) := sum((1-2^(2*k1-1)) * (-1)^(k1) * (bernoulli(2*k1)/(2*k1)) * A008956(n, n+1-k1), k1=1..n+1) / (2*4^(n)*(2*n+1)!) end do: a:=n-> numer(Delta(n)): seq(a(n), n=0..nmax-1); # _Johannes W. Meijer_, Jan 27 2009, revised Sep 21 2012
%t CoefficientList[Series[1/x - 1/Sqrt[x]/ArcSin[Sqrt[x]], {x, 0, 12}], x] // Numerator (* _Jean-François Alcover_, Jul 05 2011, after _Vladeta Jovovic_ *)
%o (Maxima)
%o a(n):=(sum(binomial(2*n+k-1,2*n-2)*sum((binomial(k+1,j)*sum((2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(n-i),i,0,j/2))/(2^(j-1)*(2*n+j)!),j,1,k+1),k,0,2*n-1))/(2*n-1);
%o makelist(num(a(n)),n,0,10); /* _Vladimir Kruchinin_, May 16 2013 */
%Y Cf. A002198.
%Y See A000367, A006954, A008956 and A002671 for underlying sequences.
%Y Factor of the LS1[-2,n] matrix coefficients in A160487.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Aug 11 2002
%E Edited by _Johannes W. Meijer_, Sep 21 2012