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A209755 Triangle of coefficients of polynomials u(n,x) jointly generated with A209756; see the Formula section. 3
1, 1, 2, 2, 4, 3, 3, 7, 8, 5, 4, 11, 17, 17, 8, 5, 16, 31, 41, 33, 13, 6, 22, 51, 83, 91, 63, 21, 7, 29, 78, 150, 205, 195, 117, 34, 8, 37, 113, 250, 406, 483, 403, 214, 55, 9, 46, 157, 392, 734, 1039, 1091, 812, 386, 89, 10, 56, 211, 586, 1239, 2023, 2536 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Column 1:  1,2,3,4,5,6,....... A000027

Column 2:  1,2,4,7,11,........ A000124

Column 3:  2,6,13,24,......... A105163

Final row terms:  1,2,3,5,.... A000045 (Fibonacci numbers)

Row sums:  1,3,9,23,57,139,... A133654

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999

For a discussion and guide to related arrays, see A208510.

LINKS

Table of n, a(n) for n=1..62.

FORMULA

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),

v(n,x)=x*u(n-1,x)+v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.

EXAMPLE

First five rows:

1

1...2

2...4....3

3...7....8....5

4...11...11...17...8

First three polynomials u(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

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

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

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[%]    (* A209755 *)

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

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

TableForm[cv]

Flatten[%]    (* A209756 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A133654 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A001333 *)

Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A033999 *)

Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A109613 *)

CROSSREFS

Cf. A209756, A208510.

Sequence in context: A128590 A143228 A143211 * A131052 A209138 A051297

Adjacent sequences:  A209752 A209753 A209754 * A209756 A209757 A209758

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 14 2012

STATUS

approved

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