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A117919 Triangle, row sums = the Pell sequence. 3
1, 1, 1, 1, 2, 2, 1, 3, 6, 2, 1, 4, 12, 8, 4, 1, 5, 20, 20, 20, 4, 1, 6, 30, 40, 60, 24, 8, 1, 7, 42, 70, 140, 84, 56, 8, 1, 8, 56, 112, 280, 224, 224, 64, 16 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row terms sums of the triangle = the Pell sequence A000129: (1, 2, 5, 12, 29...). Right border of the triangle = inverse binomial transform of the Pell sequence: (A016116): (1, 1, 2, 2, 4, 4,...).

A117919 is jointly generated with A135837 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=2x*u(n-1,x)+v(n-1,x).  See the Mathematica section. - Clark Kimberling, Feb 26 2012

LINKS

Table of n, a(n) for n=1..45.

FORMULA

The triangle = difference terms of columns from an array generated from binomial transforms of (1,0,0,0...); (1,1,0,0,0...); (1,1,2,2...); (1,1,2,2,4,...); where (1, 1, 2, 2, 4, 4,...) = A016116, the inverse binomial transform of the Pell sequence A000129..

Triangle read by rows, iterates of X * [1,0,0,0,...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (1,2,1,2,1,2,...) in the subdiagonal, with the rest zeros. - Gary W. Adamson, May 10 2008

EXAMPLE

First few rows of the generating array are:

1, 1, 1, 1, 1,...

1, 2, 3, 4, 5,...

1, 2, 5, 10, 17,...

1, 2, 5, 12, 25,...

1, 2, 5, 12, 29,...

...

Taking difference terms of the columns, we get the triangle A117919. First few rows are:

1;

1, 1;

1, 2, 2;

1, 3, 6, 2;

1, 4, 12, 8, 4;

1, 5, 20, 20, 20, 4;

1, 6, 30, 40, 60, 24, 8;

1, 7, 42, 70, 140, 84, 56, 8;

...

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x];

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]    (* A117919 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A135837 *)

CROSSREFS

Cf. A000129, A016116, A135837.

Sequence in context: A309495 A080955 A125231 * A309106 A160014 A068956

Adjacent sequences:  A117916 A117917 A117918 * A117920 A117921 A117922

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Apr 02 2006

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)