A233316 -id:A233316

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A233565

Numerators of the autosequence preceding Br(n)=A229979(n)/(1 followed by A050932(n)).

+10
1

0, 0, 0, 1, 2, 5, 5, 7, 7, 5, 5, 11, 11, 91, 91, -9, -9, 1207, 1207, -10849, -10849, 65879, 65879, -783127, -783127, 61098739, 61098739, -2034290233, -2034290233, 72986324461, 72986324461

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Br(n)=0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210, 0,.. .

a(n) is the numerators of Bp2(n)=0, 0, 0, 1, 2, 5/2, 5/2, 7/3, 7/3, 5/2, 5/2, 11/5, 11/5, 91/30, 91/30,... . Bp2(n) is an autosequence like Br(n).

With possible future sequences we can write the array PB

1, 0, 0, 0, 0, 0, 0, 0, 0,

1, 1, 0, 0, 0, 0, 0, 0, 0,

1, 3/2, 1, 0, 0, 0, 0, 0, 0,

1, 5/3, 2, 1, 0, 0, 0, 0, 0,

1, 5/3, 5/2, 5/2, 1, 0, 0, 0, 0,

1, 49/30, 5/2, 7/2, 3, 1, 0, 0, 0,

1, 49/30, 7/3, 7/2, 14/3, 7/2, 1, 0, 0,

1, 58/35, 7/3, 3, 14/3, 6, 4, 1, 0,

1, 58/35, 5/2, 3, 7/2, 6, 15/2, 9/2, 1, etc.

The first column is A000012. The second A165142(n+1)/(1 followed by A100650(n)). The third is Bp2(n+1). The next others are built by the same way. From the second,every column is based on A164555(n)/A027642(n).

With negative (2*n+2)-th diagonals,the array without 0's is the triangle NPB. The sum of every row is

1, 0, 1/2, -1/3, 1/3, -11/30, 11/30, -12/35, 12/35, -79/210, 79/210,... .

See A176250(n+2)/A100650(n).

The inverse of NPB is A193815(n)/(A003056(n) with 1 instead of 0).

LINKS

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

EXAMPLE

a(0)=a(1)=0, a(i)=numerators of 0+Br(0)=0, 0+Br(1)=1, 1+Br(2)=2, 2+Br(3)=5/2, 5/2+Br(4)=5/2,... .

MATHEMATICA

nmax = 30; Br[0] = 0; Br[1] = Br[2] = 1; Br[n_] := Numerator[2*n*BernoulliB[n-1]] / Denominator[n*BernoulliB[n-1]]; Bp2 = Join[{0, 0}, Table[Br[n], {n, 0, nmax-2}] // Accumulate]; a[n_] := Numerator[Bp2[[n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 18 2013 *)

CROSSREFS

Cf. A233316.

KEYWORD

sign

AUTHOR

Paul Curtz, Dec 13 2013

EXTENSIONS

a(17)-a(30) from Jean-François Alcover, Dec 18 2013

STATUS

approved

A233548

1, 1, followed by denominators of A000217(n)*A164555(n)/A027642(n).

+10
0

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 30, 1, 1, 1, 10, 1, 21, 1, 10, 1, 1, 1, 42, 1, 1, 1, 2, 1, 231, 1, 10, 1, 1, 1, 2730, 1, 1, 1, 110, 1, 21, 1, 2, 1, 1, 1, 1326, 1, 11, 1, 10, 1, 57, 1, 10, 1, 1, 1, 30030

(list; graph; refs; listen; history; text; internal format)

OFFSET

-2,4

COMMENTS

See A233316.

LINKS

Table of n, a(n) for n=-2..60.

MATHEMATICA

b[-2] = b[-1] = 0; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := (n + 1)*(n + 2)/2*b[n] // Denominator; Table[a[n], {n, -2, 60}]

CROSSREFS

Cf. A233316 (numerators).

KEYWORD

nonn,frac

AUTHOR

Jean-François Alcover and Paul Curtz, Dec 12 2013

STATUS

approved

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