A287703 Triangle read by rows, numerators of T(n,k) = (-1)^n*binomial(n-1,k)*Bernoulli(n+k)/ (n+k) for n>=1, 0<=k<=n-1.

1, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 5, 0, 0, 1, 0, -5, 0, 691, 0, -1, 0, 7, 0, -691, 0, 7, 0, 0, -2, 0, 691, 0, -14, 0, 3617, 0, 1, 0, -691, 0, 21, 0, -25319, 0, 43867, 0, 0, 691, 0, -10, 0, 75957, 0, -438670, 0, 174611, 0

Offset

1,20

Comments

For the rational triangle the reciprocals of the row sums are the Apéry numbers A005430.

Links

Table of n, a(n) for n=1..66.

Formula

A005430(n) = 1 / (Sum_{k=0..n-1} T(n,k)) for n>=1.

Example

The rational triangle starts (with row sums at the end of the line):
1: [1/2], 1/2
2: [1/12, 0], 1/12
3: [0, 1/60, 0], 1/60
4: [-1/120, 0, 1/84, 0], 1/280
5: [0, -1/63, 0, 1/60, 0], 1/1260
6: [1/252, 0, -1/24, 0, 5/132, 0], 1/5544
7: [0, 1/40, 0, -5/33, 0, 691/5460, 0], 1/24024
8: [-1/240, 0, 7/44, 0, -691/936, 0, 7/12, 0], 1/102960
9: [0, -2/33, 0, 691/585, 0, -14/3, 0, 3617/1020, 0], 1/437580
The numerators of the triangle are:
1: [ 1]
2: [ 1,  0]
3: [ 0,  1,  0]
4: [-1,  0,  1,   0]
5: [ 0, -1,  0,   1,    0]
6: [ 1,  0, -1,   0,    5,   0]
7: [ 0,  1,  0,  -5,    0, 691, 0]
8: [-1,  0,  7,   0, -691,   0, 7,   0]
9: [ 0, -2,  0, 691,    0, -14, 0, 3617, 0]

Maple

T := (n, k) -> numer((-1)^n*binomial(n-1, k)*bernoulli(k+n)/(k+n)):
for n from 1 to 9 do seq(T(n, k), k=0..n-1) od;

Mathematica

T[n_, k_]:=Numerator[(-1)^n*Binomial[n - 1, k] BernoulliB[k + n]/(k + n)]; Table[T[n, k], {n, 11}, {k, 0, n - 1}]//Flatten (* Indranil Ghosh, Jul 27 2017 *)

Prog

(PARI) T(n, k) = numerator((-1)^n*binomial(n-1, k)*bernfrac(k+n)/(k+n));
tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 28 2017

Crossrefs

Cf. A005430 (Apéry), A287704 (denominators).

Sequence in context: A186716 A331039 A171915 * A316480 A099224 A136598

Adjacent sequences: A287700 A287701 A287702 * A287704 A287705 A287706

Keyword

sign,tabl,frac

Author

Peter Luschny, Jun 21 2017

Status

approved