A130668 Diagonal of A129819.

0, 0, 1, -2, 5, -11, 23, -48, 102, -220, 476, -1024, 2184, -4624, 9744, -20480, 42976, -90048, 188352, -393216, 819328, -1704192, 3539200, -7340032, 15203840, -31456256, 65010688, -134217728, 276826112, -570429440, 1174409216

Offset: 0

Author

Paul Curtz, Jun 27 2007

Keywords

• sign

Comments

This sequence is connected to A124072. To see this, change the sign of every negative term and consider the differences of every line. Hence for the second line, and following lines, the four terms form periodic sequences:

0 1 0 1 0

1 0 0 1 1

1 0 1 2 1

1 1 3 3 1

2 4 6 4 2

6 10 10 6 6

16 20 16 12 16

36 36 28 28 36

72 64 56 64 72

136 120 120 136 136

256 240 256 272 256.

The lines are connected as seen by the examples: (3rd line connected to 2nd, from right to left) 1+1=2, 1+0=1, 0+0=0, 0+1=1; (11th line connected to 10th) 136+136=272, 136+120=256, 120+120=240, 120+136=256.

The 4 columns are almost known (must the first line be suppressed?): A038503 (without the first 1), A000749 (without the first 0), A038505, A038504. Like the present sequence, every sequence of A124072 beginning with a negative number (-2, -11, ...) is a "twisted" sequence (see A129339 comments, A129961 and the present 4 columns). Periodic with period 2^n.

Inverse binomial transform of A129819. - R. J. Mathar, Feb 25 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-6,-14,-16,-8).

Programs

GAP

a:=[-2,5,-11,23];; for n in [5..30] do a[n]:=-6*a[n-1]+-14*a[n-2] -16*a[n-3]-8*a[n-4]; od; Concatenation([0,0,1], a); # G. C. Greubel, Mar 24 2019

Magma

I:=[-2,5,-11,23]; [0,0,1] cat [n le 4 select I[n] else -6*Self(n-1) - 14*Self(n-2)-16*Self(n-3)-8*Self(n-4): n in [1..30]]; // G. C. Greubel, Mar 24 2019

Mathematica

gf = x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2); CoefficientList[Series[gf, {x, 0, 30}], x] (* Jean-François Alcover, Dec 16 2014, after R. J. Mathar *)
Join[{0, 0, 1}, LinearRecurrence[{-6,-14,-16,-8}, {-2,5,-11,23}, 30]] (* Jean-François Alcover, Feb 15 2016 *)

PARI

my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1+x)*(1+3*x+4*x^2+3*x^3 )/((1+2*x +2*x^2)*(1+2*x)^2))) \\ G. C. Greubel, Mar 24 2019

Sage

(x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2 )).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Mar 24 2019

Formula

From R. J. Mathar, Feb 25 2009: (Start)

G.f.: x^2*(1+x)*(1 + 3*x + 4*x^2 + 3*x^3)/((1 + 2*x + 2*x^2)*(1+2*x)^2).

a(n) = ((-1)^n*A001787(n+1) - 4*A108520(n) + 4*A122803(n))/32, n > 2. (End)

a(n) = -6*a(n-1) - 14*a(n-2) - 16*a(n-3) - 8*a(n-4) for n >= 7. - G. C. Greubel, Mar 24 2019

Extensions

Extended by R. J. Mathar, Feb 25 2009