A260196 1, -3, followed by -1's.

1, -3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Author

Paul Curtz, Jul 19 2015

Keywords

• sign
• easy

Comments

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

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.

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

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).

Crossrefs

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

Programs

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

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)