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A209996 Triangle of coefficients of polynomials u(n,x) jointly generated with A209998; see the Formula section. 3
1, 1, 3, 1, 5, 9, 1, 5, 21, 27, 1, 5, 25, 81, 81, 1, 5, 25, 117, 297, 243, 1, 5, 25, 125, 513, 1053, 729, 1, 5, 25, 125, 609, 2133, 3645, 2187, 1, 5, 25, 125, 625, 2853, 8505, 12393, 6561, 1, 5, 25, 125, 625, 3093, 12825, 32805, 41553, 19683, 1, 5, 25, 125 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n starts with 1, 5, 5^2, 5^3,...,5^floor[(n+1)/2] and

ends with 3^(n-1).  Denoting the general term by T(n,k),

we have T(n,n-1)=A081038.

Alternating row sums: A000975 (signed)

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

LINKS

Table of n, a(n) for n=1..59.

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...9

1...5...21...27

1...5...25...81...81

First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 9x^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] + 2 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[%]    (* A209996 *)

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

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

TableForm[cv]

Flatten[%]    (* A209998 *)

CROSSREFS

Cf. A209998, A208510.

Sequence in context: A094353 A298662 A306780 * A129801 A240222 A193702

Adjacent sequences:  A209993 A209994 A209995 * A209997 A209998 A209999

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 23 2012

STATUS

approved

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)