A004193 a(n) = -(-1)^n*2*(2*n+1)!*Bernoulli(2*n)/(n!*2^n).

1, 1, 5, 63, 1575, 68409, 4729725, 488783295, 71982456975, 14550187083705, 3916321542458325, 1368981608178405375, 608576219802039864375, 337967570725260384533625, 230885276313275432674678125, 191452972504088518574149173375, 190442238700388913304502070009375

Offset: 1

Author

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

Keywords

• nonn
• nice

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..100
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Bernoulli numbers at even indices are A000367/A002445.

Programs

Maple

a:= n-> -(-1)^n*2*(2*n+1)!*bernoulli(2*n)/(n!*2^n):
seq(a(n), n=1..20);  # Alois P. Heinz, Jun 13 2016

Mathematica

Table[-((-1)^n 2(2n+1)!BernoulliB[2n])/(n! 2^n),{n,20}] (* Harvey P. Dale, Oct 05 2012 *)
Table[2 (2n+1)!! Abs@BernoulliB[2n], {n, 20}] (* Vladimir Reshetnikov, Jun 05 2016 *)

PARI

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

Formula

a(n) ~ 16 * 2^(n+1/2) * Pi^(1/2-2*n) * n^(3/2) * (n/e)^(3*n). - Vladimir Reshetnikov, Sep 05 2016

From Peter Luschny, May 17 2018: (Start)

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

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

a(n) = -(-2)^(n+3)*n*Zeta(1-2*n)*(n+1/2)!/sqrt(Pi). - Peter Luschny, Jun 21 2020