cp's OEIS Frontend

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

%I A245602 #63 Dec 19 2015 22:31:48
%S A245602 -1,-3,-7,-6,-15,-60,-31,-390,-120,-63,-2100,-2520,-127,-10206,-31920,
%T A245602 -5040,-255,-46620,-317520,-181440,-511,-204630,-2739240,-3780000,
%U A245602 -362880,-1023,-874500,-21538440,-59875200,-19958400,-2047
%N A245602 Triangle read by rows: the negative terms of A163626.
%C A245602 These numbers a(n) are the companion of A249163(n).
%C A245602 Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):
%C A245602 1,
%C A245602 1, -1/2,
%C A245602 1, -3/2,  +2/3,
%C A245602 1, -7/2, +12/3, -6/4,
%C A245602 etc.
%C A245602 From the second row on, the sum of the numerators is 0.
%C A245602 The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).
%C A245602 a(n) triangle is shifted. It starts from second row and second column of triangle above.
%C A245602 -1,
%C A245602 -3,
%C A245602 -7,       -6,
%C A245602 -15,     -60,
%C A245602 -31,    -390,    -120,
%C A245602 -63,   -2100,   -2520,
%C A245602 -127, -10206,  -31920,   -5040,
%C A245602 -255, -46620, -317520, -181440,
%C A245602 etc.
%C A245602 Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).
%C A245602 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 A245602 Thanks to _Jean-François Alcover_.
%t A245602 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 A245602 Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.
%K A245602 sign,tabf
%O A245602 0,2
%A A245602 _Paul Curtz_, Dec 17 2014