A004193 - OEIS

%I #36 Jun 21 2020 06:02:28

%S 1,1,5,63,1575,68409,4729725,488783295,71982456975,14550187083705,

%T 3916321542458325,1368981608178405375,608576219802039864375,

%U 337967570725260384533625,230885276313275432674678125,191452972504088518574149173375,190442238700388913304502070009375

%N a(n) = -(-1)^n*2*(2*n+1)!*Bernoulli(2*n)/(n!*2^n).

%D J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 35, Eq. 4:2:1.

%H G. C. Greubel, <a href="/A004193/b004193.txt">Table of n, a(n) for n = 1..100</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%F a(n) ~ 16 * 2^(n+1/2) * Pi^(1/2-2*n) * n^(3/2) * (n/e)^(3*n). - _Vladimir Reshetnikov_, Sep 05 2016

%F From _Peter Luschny_, May 17 2018: (Start)

%F a(n) ~ 8*sqrt(2*n*Pi)*(2*Pi)^n*(n/(Pi*e))^(3*n)*(2*n+1).

%F a(n) = |(2^(n+2)*Pochhammer(1/2, n+1)*Bernoulli(2*n)|. (End)

%F a(n) = -(-2)^(n+3)*n*Zeta(1-2*n)*(n+1/2)!/sqrt(Pi). - _Peter Luschny_, Jun 21 2020

%p a:= n-> -(-1)^n*2*(2*n+1)!*bernoulli(2*n)/(n!*2^n):

%p seq(a(n), n=1..20);  # _Alois P. Heinz_, Jun 13 2016

%t Table[-((-1)^n 2(2n+1)!BernoulliB[2n])/(n! 2^n),{n,20}] (* _Harvey P. Dale_, Oct 05 2012 *)

%t Table[2 (2n+1)!! Abs@BernoulliB[2n], {n, 20}] (* _Vladimir Reshetnikov_, Jun 05 2016 *)

%o (PARI) a(n)=if(n<1,0,-(-1)^n*2*(2*n+1)!*bernfrac(2*n)/(n!*2^n))

%Y Bernoulli numbers at even indices are A000367/A002445.

%K nonn,nice

%O 1,3

%A David W. Cantrell (DWCantrell(AT)sigmaxi.net)