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A130668 Diagonal of A129819. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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).

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

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 *)

PROG

(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

(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

(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

(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

CROSSREFS

Sequence in context: A292937 A034468 A283075 * A358912 A083380 A018112

Adjacent sequences: A130665 A130666 A130667 * A130669 A130670 A130671

KEYWORD

sign

AUTHOR

Paul Curtz, Jun 27 2007

EXTENSIONS

Extended by R. J. Mathar, Feb 25 2009

STATUS

approved

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Last modified February 1 14:40 EST 2023. Contains 359993 sequences. (Running on oeis4.)