A002195 Numerators of coefficients for numerical integration. (Formerly M4809 N2056)

1, -1, 11, -191, 2497, -14797, 92427157, -36740617, 61430943169, -23133945892303, 16399688681447, -3098811853954483, 312017413700271173731, -69213549869569446541, 53903636903066465730877, -522273861988577772410712439, 644962185719868974672135609261

Offset

0,3

Comments

The numerators 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 numerators of the constant term in the Laurent expansion of the 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..16.

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) = numerator of (2/(2*n+1)!)*Integral_{t=0..1} t*Product_{k=1..n} t^2-k^2. - Emeric Deutsch, Jan 25 2005

a(0) = 1; a(n) = numerator [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

a(n) = numerator(sum(k=1..2*n, binomial(2*n+k-1,2*n-1)*sum(j=1..k, (binomial(k,j)*sum(i=0,j/2, (2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(-i)))/(2^j*(2*n+j)!)))), n>0, a(0)=1. - Vladimir Kruchinin, Feb 04 2013

Example

a(1) = -1 because (1/3)*int(t*(t^2-1^2),t=0..1) = -1/12.
a(3) = numer((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = -191. - Johannes W. Meijer, Jan 27 2009

Maple

a:=n->numer((2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1)): seq(a(n), n=0..16); # Emeric Deutsch, Feb 20 2005
nmax:=16: 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-> numer(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}]; Table[a[n] // Numerator, {n, 0, 16}] (* Jean-François Alcover, Apr 18 2014, after Vladimir Kruchinin *)
a[n_] := Numerator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* Istvan Mezo, Apr 21 2023 *)

Prog

(Maxima)
a(n):=num(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)); /* Vladimir Kruchinin, Feb 04 2013 */

Crossrefs

Cf. A002196.

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

Sequence in context: A298643 A185123 A036936 * A280070 A171553 A333759

Adjacent sequences: A002192 A002193 A002194 * A002196 A002197 A002198

Keyword

sign,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