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%I M4880 N2093 #47 Jun 11 2023 12:12:01
%S 1,12,720,60480,3628800,95800320,2615348736000,4483454976000,
%T 32011868528640000,51090942171709440000,152579284313702400000,
%U 120866571766215475200000,50814724101952310083584000000
%N Denominators of coefficients for numerical integration.
%C The denominators 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 These numbers are the denominators of the constant term in the Laurent expansion of the even powers of the hyperbolic cosecant cosech^(2n)(x)/2^(2n) function. - _Istvan Mezo_, Apr 21 2023
%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 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 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 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 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 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 a(n) = denominator of (2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1). - _Emeric Deutsch_, Jan 25 2005
%F a(0) = 1; a(n) = denominator [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
%e a(1) = 12 because (1/3)*int(t*(t^2-1^2), t=0..1) = -1/12.
%e a(3) = denom((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = 60480. - _Johannes W. Meijer_, Jan 27 2009
%p 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); # _Emeric Deutsch_, Feb 20 2005
%p nmax:=12: 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-> denom(Omega(n)): seq(a(n), n=0..nmax); # _Johannes W. Meijer_, Jan 27 2009, Revised Sep 21 2012
%t 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}]/2^(2*n-1); Table[a[n] // Denominator, {n, 0, 12}] (* _Jean-François Alcover_, Apr 18 2014, after _Vladimir Kruchinin_ *)
%t a[n_] := Denominator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* _Istvan Mezo_, Apr 21 2023 *)
%Y Cf. A002195.
%Y See A000367, A006954, A008955 and A009445 for underlying sequences.
%Y Factor of ZS1[ -1,n] matrix coefficients in A160474.
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Jan 25 2005
%E Edited by _Johannes W. Meijer_, Sep 21 2012
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