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A209819 Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section. 3
1, 1, 3, 1, 5, 7, 1, 5, 17, 17, 1, 5, 21, 53, 41, 1, 5, 21, 81, 157, 99, 1, 5, 21, 89, 289, 449, 239, 1, 5, 21, 89, 361, 973, 1253, 577, 1, 5, 21, 89, 377, 1389, 3133, 3433, 1393, 1, 5, 21, 89, 377, 1565, 5085, 9745, 9273, 3363, 1, 5, 21, 89, 377, 1597, 6285 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Let T(n,k) be the general term.

T(n,n): A001333

T(n,n-1): A088210

Row sums: A003561

Alternating row sums: 1,-2,3,-4,5,-6,7,-8,...

Limiting row: F(2), F(5),F(8),...where F=A000045 (Fibonacci numbers)

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)+2x*v(n-1,x)+1,

v(n,x)=(x+1)*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...5...7

1...5...17...17

1...5...21...53...41

First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 7x^2.

MATHEMATICA

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

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

v[n_, x_] := (x + 1)*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[%]    (* A209819 *)

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

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

TableForm[cv]

Flatten[%]    (* A209820 *)

CROSSREFS

Cf. A209820, A208510.

Sequence in context: A016600 A130418 A038871 * A193648 A221881 A201811

Adjacent sequences:  A209816 A209817 A209818 * A209820 A209821 A209822

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 23 2012

STATUS

approved

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Last modified October 20 22:05 EDT 2019. Contains 328291 sequences. (Running on oeis4.)