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A209763 Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section. 3
1, 1, 2, 2, 5, 4, 3, 9, 13, 8, 4, 15, 31, 35, 16, 5, 23, 61, 97, 85, 32, 6, 33, 107, 219, 279, 203, 64, 7, 45, 173, 433, 717, 761, 469, 128, 8, 59, 263, 779, 1583, 2195, 1991, 1067, 256, 9, 75, 381, 1305, 3141, 5361, 6381, 5049, 2389, 512, 10, 93, 531 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n begins with n and ends with 2^(n-1).

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

Alternating row sums: 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..58.

FORMULA

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

v(n,x)=2x*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...5....4

3...9....13...8

4...15...31...35...16

First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^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] + 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[%]    (* A209763 *)

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

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

TableForm[cv]

Flatten[%]    (* A209764 *)

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}] (* A004442 *)

CROSSREFS

Cf. A209764, A208510.

Sequence in context: A209751 A275381 A283235 * A209761 A228526 A209745

Adjacent sequences:  A209760 A209761 A209762 * A209764 A209765 A209766

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 14 2012

STATUS

approved

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Last modified October 13 19:52 EDT 2019. Contains 327981 sequences. (Running on oeis4.)