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A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) is the numerators of numbers issued of 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.

LINKS

Table of n, a(n) for n=0..30.

FORMULA

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

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

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 *)

CROSSREFS

Cf. A083420.

Sequence in context: A253892 A116419 A290235 * A199927 A099398 A272276

Adjacent sequences:  A225822 A225823 A225824 * A225826 A225827 A225828

KEYWORD

sign

AUTHOR

Paul Curtz, Jul 30 2013

EXTENSIONS

More terms from Jean-François Alcover, Aug 01 2013

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

STATUS

approved

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Last modified October 16 08:50 EDT 2019. Contains 328056 sequences. (Running on oeis4.)