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A209765 Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section. 3
1, 1, 2, 1, 5, 5, 1, 5, 15, 12, 1, 5, 21, 45, 29, 1, 5, 21, 77, 129, 70, 1, 5, 21, 89, 265, 361, 169, 1, 5, 21, 89, 353, 865, 991, 408, 1, 5, 21, 89, 377, 1325, 2717, 2681, 985, 1, 5, 21, 89, 377, 1549, 4733, 8281, 7169, 2378, 1, 5, 21, 89, 377, 1597, 6125 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Limiting row: F(2+3k), where F=A000045 (Fibonacci numbers)

Coefficient of x^n in u(n,x): 1,2,5,12,.... A000129(n)

Row sums:  1,3,11,33,101,303,911,2733,..... A081250

Alternating row sums: 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)=2x*u(n-1,x)+x*v(n-1,x)+1,

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

EXAMPLE

First five rows:

1

1...2

1...5...5

1...5...15...12

1...5...21...45...29

First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^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_] := 2 x*u[n - 1, x] + 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[%]    (* A209765 *)

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

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

TableForm[cv]

Flatten[%]    (* A209766 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)

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

Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

CROSSREFS

Cf. A209766, A208510.

Sequence in context: A005605 A300661 A145882 * A209759 A111785 A304462

Adjacent sequences:  A209762 A209763 A209764 * A209766 A209767 A209768

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 14 2012

STATUS

approved

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)