A002196 Denominators of coefficients for numerical integration. (Formerly M4880 N2093)

1, 12, 720, 60480, 3628800, 95800320, 2615348736000, 4483454976000, 32011868528640000, 51090942171709440000, 152579284313702400000, 120866571766215475200000, 50814724101952310083584000000

Offset

0,2

Comments

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

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

References

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).

Links

Table of n, a(n) for n=0..12.

Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.

H. E. Salzer, Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264. [Annotated scanned copy]

H. E. Salzer, XXXII. 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.

T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Formula

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

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

Example

a(1) = 12 because (1/3)*int(t*(t^2-1^2), t=0..1) = -1/12.
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

Maple

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
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

Mathematica

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 *)
a[n_] := Denominator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* Istvan Mezo, Apr 21 2023 *)

Crossrefs

Cf. A002195.

See A000367, A006954, A008955 and A009445 for underlying sequences.

Factor of ZS1[ -1,n] matrix coefficients in A160474.

Sequence in context: A215686 A277691 A262383 * A141421 A000909 A358732

Adjacent sequences: A002193 A002194 A002195 * A002197 A002198 A002199

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Jan 25 2005

Edited by Johannes W. Meijer, Sep 21 2012

Status

approved