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Triangle read by rows: the negative terms of A163626.
1

%I #63 Dec 19 2015 22:31:48

%S -1,-3,-7,-6,-15,-60,-31,-390,-120,-63,-2100,-2520,-127,-10206,-31920,

%T -5040,-255,-46620,-317520,-181440,-511,-204630,-2739240,-3780000,

%U -362880,-1023,-874500,-21538440,-59875200,-19958400,-2047

%N Triangle read by rows: the negative terms of A163626.

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

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

%C 1,

%C 1, -1/2,

%C 1, -3/2, +2/3,

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

%C etc.

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

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

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

%C -1,

%C -3,

%C -7, -6,

%C -15, -60,

%C -31, -390, -120,

%C -63, -2100, -2520,

%C -127, -10206, -31920, -5040,

%C -255, -46620, -317520, -181440,

%C etc.

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

%C 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.

%C Thanks to _Jean-François Alcover_.

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

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

%K sign,tabf

%O 0,2

%A _Paul Curtz_, Dec 17 2014