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A117938 Triangle, columns generated from Lucas Polynomials. 5
1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 11, 14, 7, 1, 5, 18, 36, 34, 11, 1, 6, 27, 76, 119, 82, 18, 1, 7, 38, 140, 322, 393, 198, 29, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3665, 3778, 4287, 1154, 76 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).

A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019

LINKS

Table of n, a(n) for n=1..55.

FORMULA

Columns are f(x), x = 1,2,3..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3x); (x^4 + 4x^2 + 2); (x^5 + 5x^3 + 5x); (x^6 + 6x^4 + 9x^2 + 2); (x^7 + 7x^5 + 14x^3 + 7x);...

EXAMPLE

First few rows of the triangle are:

1;

1, 1;

1, 2, 3;

1, 3, 6, 4;

1, 4, 11, 14, 7;

1, 5, 18, 36, 34, 11;

1, 6, 27, 76, 119, 82, 18;

1, 7, 38, 140, 322, 393, 198, 29;

...

For example, T(7,4) = 76 = f(4), x^3 + 3x = 64 + 12 = 76.

MAPLE

Lucas := proc(n, x) # see A114525

    option remember;

    if  n=0 then

        2;

    elif n =1 then

        x ;

    else

        x*procname(n-1, x)+procname(n-2, x) ;

    end if;

    expand(%) ;

end proc:

A117938 := proc(n::integer, k::integer)

    if k = 1 then

        1;

    else

        subs(x=n-k+1, Lucas(k-1, x)) ;

    end if;

end proc:

seq(seq(A117938(n, k), k=1..n), n=1..12) ; # R. J. Mathar, Aug 16 2019

CROSSREFS

Cf. A114525, A104509, A117936, A117937, A118980, A118981, A309220, A059100 (column 3), A061989 (column 4), A000204 (diagonal).

Sequence in context: A152976 A153861 A118981 * A256193 A101912 A208522

Adjacent sequences:  A117935 A117936 A117937 * A117939 A117940 A117941

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, Apr 03 2006

STATUS

approved

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Last modified May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)