A245602 Triangle read by rows: the negative terms of A163626.

-1, -3, -7, -6, -15, -60, -31, -390, -120, -63, -2100, -2520, -127, -10206, -31920, -5040, -255, -46620, -317520, -181440, -511, -204630, -2739240, -3780000, -362880, -1023, -874500, -21538440, -59875200, -19958400, -2047

Offset: 0

Author

Paul Curtz, Dec 17 2014

Keywords

• sign
• tabf

Comments

These numbers a(n) are the companion of A249163(n).

Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):

1,

1, -1/2,

1, -3/2, +2/3,

1, -7/2, +12/3, -6/4,

etc.

From the second row on, the sum of the numerators is 0.

The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).

a(n) triangle is shifted. It starts from second row and second column of triangle above.

-1,

-3,

-7, -6,

-15, -60,

-31, -390, -120,

-63, -2100, -2520,

-127, -10206, -31920, -5040,

-255, -46620, -317520, -181440,

etc.

Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).

Successive columns: A000225, A028244, from the Stirling numbers of second kind S(n,2), S(n,4), S(n,6), S(n,8), S(n,10), ... . See A000770, A032180, A049434, A228910, A049435, A228912, A008277.

Thanks to Jean-François Alcover.

Crossrefs

Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.

Programs

Mathematica

Select[ Table[ (-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten, Negative] (* Jean-François Alcover, Dec 26 2014 *)