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A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section. 3
1, 1, 3, 1, 3, 7, 1, 3, 13, 17, 1, 3, 13, 43, 41, 1, 3, 13, 55, 133, 99, 1, 3, 13, 55, 209, 391, 239, 1, 3, 13, 55, 233, 739, 1113, 577, 1, 3, 13, 55, 233, 939, 2469, 3095, 1393, 1, 3, 13, 55, 233, 987, 3589, 7903, 8457, 3363, 1, 3, 13, 55, 233, 987, 4085 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Limiting row: F(1+3k), where F=A000045 (Fibonacci numbers)

Coefficient of x^n in u(n,x): A001333(n)

Row sums:  1,4,11,34,101,304,... A060925.

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...3

1...3...7

1...3...13...17

1...3...13...43...41

First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 7x^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. A209665, A208510.

Sequence in context: A208517 A209566 A208916 * A114972 A107461 A035619

Adjacent sequences:  A209763 A209764 A209765 * A209767 A209768 A209769

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 14 2012

STATUS

approved

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Last modified October 16 01:33 EDT 2019. Contains 328038 sequences. (Running on oeis4.)