A176150 Numerators of the inverse binomial transform of a shuffled sequence of "original" Bernoulli and Bernoulli numbers.

1, 0, -1, 0, 13, -20, 14, -73, 373, -776, 196, -223, 1027, -1956, 242, 139, -3521, 2288, -824, -36433, 600283, -502708, 83638, -40071, 4098793, 1096584, -1160948, 3484553, -6256069, 4889852, -1520674, -276156781

Offset: 0

Author

Paul Curtz, Apr 10 2010, Apr 19 2010

Keywords

• frac
• sign
• uned

Comments

Define a shuffled sequence bb1(n) by taking the "original" Bernoulli numbers for even indices and the Bernoulli numbers for odd indices: bb1(2n) = A164555(n)/A027642(n), bb1(2n+1) = A027641(n)/A027642(n).

This starts bb1(n) = 1, 1, 1/2, -1/2, 1/6, 1/6, 0, 0, -1/30, -1/30, 0, 0, 1/42, 1/42,...

The inverse binomial transform of bb1(n) starts 1, 0, -1/2, 0, 13/6, -20/3, 14, -73/3 and its numerators define the current sequence.

Programs

Maple

bb1 := proc(n)
    nh := floor(n/2) ;
    if n <= 1 then
        bernoulli(nh) ;
    elif n <= 3 then
        -(-1)^n*bernoulli(nh) ;
    else
        bernoulli(nh) ;
    end if;
end proc:
[seq(bb1(n),n=0..40)] ;
read("transforms");
BINOMIALi(%) ;
numer(%) ; # R. J. Mathar, Mar 26 2013