A233316 a(n) = Numerator(binomial(n+2, 2)*Bernoulli(n, 1)) for n >= 0 and 0 for n < 0.

0, 0, 1, 3, 1, 0, -1, 0, 2, 0, -3, 0, 5, 0, -691, 0, 140, 0, -10851, 0, 219335, 0, -1222277, 0, 1709026, 0, -1181820455, 0, 538845489, 0, -23749461029, 0, 68926730208040, 0, -84802531453387, 0, 270657225128535, 0, -26315271553053477373, 0, 380899208799402670, 0, -1827579029475143854357

Offset

-2,4

Comments

Numerators of 0, 0, followed by A000217(n)*A164555(n)/A027642(n).

Third fractional autosequence after (1) (Br0(n) = ) A164555/A027642 and (2) Br(n) = A229979/(c(n) = 1,1,1,2,1,6,... = 1 interleaved with A006955 or 1 followed by A050932; thanks to Jean-François Alcover). Hence

Br2(n) = 0, 0, 1, 3/2, 1, 0, -1/2, 0, 2/3, 0, -3/2, 0, 5, 0, -691/30, ..., second complementary Bernoulli numbers.

Br2(n) differences table:

0, 0, 1, 3/2, 1, 0, -1/2, ...

0, 1, 1/2, -1/2, -1, -1/2, 1/2, ...

1, -1/2, -1, -1/2, 1/2, 1, 1/6, ...

-3/2, -1/2, 1/2, 1, 1/2, -5/6, -3/2, ...

1, 1, 1/2, -1/2, -4/3, -2/3, 2, ...

0, -1/2, -1, -5/6, 2/3, 8/3, 4/3, ...

-1/2, -1/2, 1/6, 3/2, 2, -4/3, -8, ... .

The main diagonal is the double of the first upper diagonal. Then, the autosequence (its inverse binomial transform is the signed sequence) is of second kind. Note that Br0(n) is an autosequence of second kind and Br(n) an autosequence of first kind.

First Bernoulli polynomials, i.e., for B(1) = -1/2, A196838/A196839, with 0's instead of the spaces:

1, 0, 0, 0, 0, 0, 0, 0, 0, ...

-1/2, 1, 0, 0, 0, 0, 0, 0, 0, ...

1/6, -1, 1, 0, 0, 0, 0, 0, 0, ...

0, 1/2, -3/2, 1, 0, 0, 0, 0, 0, ...

-1/30, 0, 1, -2, 1, 0, 0, 0, 0, ...

0, -1/6, 0, 5/3, -5/2, 1, 0, 0, 0, ...

1/42, 0, -1/2, 0, 5/2, -3, 1, 0, 0, ...

0, 1/6, 0, -7/6, 0, 0, -7/2, 1, 0, ...

-1/30, 0, 2/3, 0, -7/3, 0, 14/3, -4, 10, ... .

First column: A164555/A027642 with -1/2 instead of 1/2, A027641/A027642.

Second column: A229979/c(n) with -1 instead of 1, first column in A229979.

Third column: Br2(n) with -3/2 instead of 3/2, first column of the first array.

Etc.

Sequences used for Brp(n). For p=1, Br(n) is used.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... = A001477,

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, ...

0, 0, 0, 1, 4, 10, 20, 35, 56, 84, ... . See A052553.

For every sequence, the multiplication by A164555/A027642 begins at 1.

(Br0(n), Br(n), Br2(n), Br3(n), ... lead to A193815.)

Links

Table of n, a(n) for n=-2..40.

Example

a(2) =    1*1 =   1,
a(3) =  3*1/2 =  3/2,
a(4) =    6/6 =   1,
a(5) =   10*0 =   0,
a(6) = -15/30 = -1/2.

Mathematica

b[-2] = b[-1] = 0; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := (n+1)*(n+2)/2*b[n] // Numerator; Table[a[n], {n, -2, 40}] (* Jean-François Alcover, Dec 09 2013 *)

Crossrefs

Cf. A190339(array). A000012 for Br0(n), A000217 for Br(n)=A229979/c(n), A000217 for Br2(n), A000292 for Br3(n).

Sequence in context: A227570 A352269 A111700 * A060096 A245756 A360672

Adjacent sequences: A233313 A233314 A233315 * A233317 A233318 A233319

Keyword

sign,frac,tabl,uned

Author

Paul Curtz, Dec 07 2013

Extensions

Corrected and extended by Jean-François Alcover, Dec 09 2013

Status

approved