A002197 Numerators of coefficients for numerical integration. (Formerly M5049 N2183)

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

Offset

0,2

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

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A002198.

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

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

Sequence in context: A222678 A293691 A365678 * A070148 A097499 A132541

Adjacent sequences: A002194 A002195 A002196 * A002198 A002199 A002200

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Aug 11 2002

Edited by Johannes W. Meijer, Sep 21 2012

Status

approved