OFFSET
0,2
COMMENTS
For more on the Akiyama-Tanigawa transform, see Links (correction: page 7 read here A164555 instead of A027641) and A177427.
Here:
1, -3, -1, -1, -1, -1, ...
4, -4, 0, 0, 0, 0, ...
8, -8, 0, 0, 0, 0, ...
16, -16, 0, 0, 0, 0, ...
etc.
1, 1/1, 1,
1, 4, ( 1, -1)/1, -3,
1, 4, 8, ( 2, -3, 1)/2, -1,
1, 4, 8, 16, * ( 6, -11, 6, -1)/6, = -1,
1, 4, 8, 16, 32, ( 24, -50, 35, -10, 1)/24, -1,
1, 4, 8, 16, 32, 64, (120, -274, 225, -85, 15, -1)/120, -1,
etc. etc. etc.
Via the modified Stirling numbers of the first kind, the second triangle, Iw(n), is the inverse of Worpitzky transform A163626(n).
a(n) is the third sequence of a family beginning with
1, 1, 1, 1, 1, 1, 1, 1, ... = A000012(n)
1, 0, 0, 0, 0, 0, 0, 0, 0, ... = A000007(n)
1, -3, -1, -1, -1, -1, -1, -1, -1, -1, ... .
A000012(n) is the inverse Akiyama-Tanigawa transform of A000007(n), with or without its second term.
A000007(n) is the inverse Akiyama-Tanigawa transform of A000012(n), with or without its second term.
a(n) is the inverse Akiyama-Tanigawa transform of 2^n omitting the second term i.e. 2.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, Journal of Integer Sequences, 3(2000), article 00.2.9
Index entries for linear recurrences with constant coefficients, signature (1).
FORMULA
Inverse Akiyama-Tanigawa transform of A151821(n).
From Colin Barker, Sep 11 2015: (Start)
a(n) = -1 for n>1.
a(n) = a(n-1) for n>2.
G.f.: -(2*x^2-4*x+1) / (x-1).
(End)
PROG
(PARI) first(m)=vector(m, i, i--; if(i>1, -1, if(i==0, 1, if(i==1, -3)))) \\ Anders Hellström, Aug 28 2015
(PARI) Vec(-(2*x^2-4*x+1)/(x-1) + O(x^100)) \\ Colin Barker, Sep 11 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Jul 19 2015
STATUS
approved