A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

1, 1, -1, -1, 7, 3, -31, -17, 127, 155, -2555, -2073, 1414477, 38227, -57337, -929569, 118518239, 28820619, -5749691557, -1109652905, 91546277357, 51943281731, -1792042792463, -2905151042481, 1982765468311237, 191329672483963, -286994504449393, -14655626154768697, 3187598676787461083, 1291885088448017715, -4625594554880206790555

Offset: 0

Author

Paul Curtz, Jul 30 2013

Keywords

• sign

Comments

a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.

The numbers are

BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..

Difference table:

1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42,...

-1/2, -2/3, -1/3, 11/15, 19/15, -47/21, -163/21,...

-1/6, 1/3, 16/15, 8/15, -368/105, -116/21, 2152/105,...

1/2, 11/15, -8/15, -424/105, -212/105, 2732/105, 4204/105,...

7/30, -19/15, -368/195, 212/105, 2944/105, 1472/105,...

-3/2, -47/21, 116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .

a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.

a(n) is divisible by A051716(n+1).

Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.

The numerators of the main diagonal:

1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...

(thanks to Jean-François Alcover) are divisible by 2^n.

Crossrefs

Cf. A083420.

Programs

Maple

A225825 := proc(n)
    local nhalf ;
    nhalf := floor(n/2) ;
    if type(n,'even') then
        A001896(nhalf) ;
    else
        (-1)^nhalf*A110501(nhalf+1) ;
    end if;
end proc; # R. J. Mathar, Oct 28 2013

Mathematica

a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 01 2013 *)

Formula

c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .

a(n) = A157779(n) + c(n).

Extensions

More terms from Jean-François Alcover, Aug 01 2013

Definition corrected by R. J. Mathar, Oct 28 2013