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A180165 Triangle read by rows, derived from an array of sequences generated from (1 + x)/ (1 - r*x - r*x^2) 3
1, 1, 2, 1, 3, 3, 1, 4, 8, 5, 1, 5, 15, 22, 8, 1, 6, 24, 57, 60, 13, 1, 7, 35, 116, 216, 164, 21, 1, 8, 48, 205, 560, 819, 488, 34, 1, 9, 63, 330, 1200, 2704, 3105, 1224, 55, 1, 10, 80, 497, 2268, 7025, 13056, 11772, 3344, 89, 1, 11, 99, 712, 3920, 15588, 41125, 63040 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums = A180166: (1, 3, 7, 18, 51, 161, 560, 2163,...).

Rows of the array, with other offsets: (row 1 = A000045 starting with offset 2: (1, 2, 3, 5, 8, 13,...); and for rows >1, the entries: A028859, A125145, A086347, and A180033 start with offset 0; with the offset in the present array = 1.

LINKS

Table of n, a(n) for n=1..63.

FORMULA

Triangle read by rows, generated from an array of sequences generated from (1 + x)/(1 - r*x - r*x^2); r>0.

Alternatively, given the array with offset 1, the sequence r-th sequence is generated from a(k) = r*a(k-1) + r*(k-2); a(1) = 1, a(2) = (r+1).

With a matrix method, the array can be generated from a 2x2 matrix of the form [0,1; r,r] = M, such that M^n * [1,(r+1] = [(r,(n+1); r,(n+2)].

Also, (r+1)-th row of the array, r>1 = the INVERT transform of r-th row.

EXAMPLE

First few rows of the triangle =

.

1;

1, 2;

1, 3, 3;

1, 4, 8, 5;

1, 5, 15, 22, 8;

1, 6, 24, 57, 60, 13;

1, 7, 35, 116, 216, 164, 21;

1, 8, 48, 205, 560, 819, 488, 34;

1, 9, 63, 330, 1200, 2704, 3105, 1224, 55;

1, 10, 80, 497, 2268, 7025, 13056, 11772, 3344, 89;

1, 11, 99, 712, 3920, 15588, 41125, 63040, 44331, 9136, 144;

1, 12, 120, 981, 6336, 30919, 107136, 240750, 304384, 169209, 24960, 233;

...

Examples: Row 5 = A180033 = (1, 6, 35, 205,...) and is generated from (1+x)/(1-5*x-5*x^2); is the INVERT transform of row 4; and the array term (5,4) = 205 = 5*35 + 5*6.

Terms (2,4) and (2,5) = [22,60] = [0,1; 2,2]^3 * [1,3].

CROSSREFS

Cf. A180166, A000045, A028859, A125145, A086347, A180033

Sequence in context: A208597 A179943 A089944 * A142249 A274705 A257243

Adjacent sequences:  A180162 A180163 A180164 * A180166 A180167 A180168

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 14 2010

STATUS

approved

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Last modified June 2 11:35 EDT 2020. Contains 334771 sequences. (Running on oeis4.)