A260196 - OEIS

%I #42 Dec 24 2018 03:25:30

%S 1,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,

%T -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,

%U -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1

%N 1, -3, followed by -1's.

%C 1/(n+1) is the inverse Akiyama-Tanigawa transform of A164555(n)/A027642(n).

%C For more on the Akiyama-Tanigawa transform, see Links (correction: page 7 read here A164555 instead of A027641) and A177427.

%C Here:

%C    1,  -3, -1, -1, -1, -1, ...

%C    4,  -4,  0,  0,  0,  0, ...

%C    8,  -8,  0,  0,  0,  0, ...

%C   16, -16,  0,  0,  0,  0, ...

%C   etc.

%C Other process, using signed A130534(n), different of A008275(n):

%C 1,                            1/1,                                      1,

%C 1, 4,                      (  1,     -1)/1,                            -3,

%C 1, 4, 8,                   (  2,     -3,   1)/2,                       -1,

%C 1, 4, 8, 16,          *    (  6,    -11,   6,   -1)/6,             =   -1,

%C 1, 4, 8, 16, 32,           ( 24,    -50,  35,  -10,  1)/24,            -1,

%C 1, 4, 8, 16, 32, 64,       (120,   -274, 225,  -85, 15, -1)/120,       -1,

%C etc.                       etc.                                        etc.

%C Via the modified Stirling numbers of the first kind, the second triangle, Iw(n), is the inverse of Worpitzky transform A163626(n).

%C a(n) is the third sequence of a family beginning with

%C 1,  1,  1,  1,  1,  1,  1,  1, ...              = A000012(n)

%C 1,  0,  0,  0,  0,  0,  0,  0,  0, ...          = A000007(n)

%C 1, -3, -1, -1, -1, -1, -1, -1, -1, -1, ... .

%C A000012(n) is the inverse Akiyama-Tanigawa transform of A000007(n), with or without its second term.

%C A000007(n) is the inverse Akiyama-Tanigawa transform of A000012(n), with or without its second term.

%C a(n) is the inverse Akiyama-Tanigawa transform of 2^n omitting the second term i.e. 2.

%H Colin Barker, <a href="/A260196/b260196.txt">Table of n, a(n) for n = 0..1000</a>

%H Masanobu Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, Journal of Integer Sequences, 3(2000), article 00.2.9

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F Inverse Akiyama-Tanigawa transform of A151821(n).

%F From _Colin Barker_, Sep 11 2015: (Start)

%F a(n) = -1 for n>1.

%F a(n) = a(n-1) for n>2.

%F G.f.: -(2*x^2-4*x+1) / (x-1).

%F (End)

%o (PARI) first(m)=vector(m,i,i--;if(i>1,-1,if(i==0,1,if(i==1,-3)))) \\ _Anders Hellström_, Aug 28 2015

%o (PARI) Vec(-(2*x^2-4*x+1)/(x-1) + O(x^100)) \\ _Colin Barker_, Sep 11 2015

%Y Cf. A000007, A000012, A008275, A027642, A130534, A151821, A163626, A164555, A177427.

%K sign,easy

%O 0,2

%A _Paul Curtz_, Jul 19 2015