A002197 Numerators of coefficients for numerical integration.

1, 17, 367, 27859, 1295803, 5329242827, 25198857127, 11959712166949, 11153239773419941, 31326450596954510807, 3737565567167418110609, 2102602044094540855003573, 189861334343507894443216783

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			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
		

References

• H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
• 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

• T. D. Noe, Table of n, a(n) for n=0..100
• H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
• T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Crossrefs

Cf. A002198.

See A000367, A006954, A008956 and A002671 for underlying sequences.

Factor of the LS1[-2,n] matrix coefficients in A160487.

Programs

Maple

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

Mathematica

CoefficientList[Series[1/x - 1/Sqrt[x]/ArcSin[Sqrt[x]], {x, 0, 12}], x] // Numerator (* Jean-François Alcover, Jul 05 2011, after Vladeta Jovovic *)

Maxima

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);
makelist(num(a(n)),n,0,10); /* Vladimir Kruchinin, May 16 2013 */

Formula

Numerators of coefficients in expansion of 1/x-1/sqrt(x)/arcsin(sqrt(x)). - Vladeta Jovovic, Aug 11 2002

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

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

Extensions

More terms from Vladeta Jovovic, Aug 11 2002

Edited by Johannes W. Meijer, Sep 21 2012