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A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section. 5
1, 1, 2, 2, 4, 3, 3, 9, 10, 5, 5, 18, 28, 22, 8, 8, 35, 68, 74, 45, 13, 13, 66, 154, 210, 177, 88, 21, 21, 122, 331, 541, 574, 397, 167, 34, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55, 55, 399, 1382, 2982, 4404, 4614, 3434, 1758, 566, 89, 89, 710, 2723 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Every row begins and ends with a Fibonacci number (A000045).

u(n,1) = n-th row sum = 3^(n-1).

alternating row sums: 1,-1,1,-1,1,-1,1,-1,1,-1,...

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

LINKS

Table of n, a(n) for n=1..58.

FORMULA

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

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

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

T(n,k) = A185081(n,k+1). - Philippe Deléham, Apr 11 2012

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n.

EXAMPLE

First five rows:

1

1...2

2...4....3

3...9....10...5

5...18...28...22...8

First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.

Triangle in A185081 begins :

1

0, 1

0, 1, 2

0, 2, 4, 3

0, 3, 9, 10, 5

0, 5, 18, 28, 22, 8 . Philippe Deléham, Apr 11 2012

MATHEMATICA

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

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

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

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[%]    (* A209137 *)

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

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

TableForm[cv]

Flatten[%]    (* A209138 *)

CROSSREFS

Cf. A209137, A208510.

Sequence in context: A143211 A209755 A131052 * A051297 A078317 A105016

Adjacent sequences:  A209135 A209136 A209137 * A209139 A209140 A209141

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 19 11:26 EDT 2019. Contains 328216 sequences. (Running on oeis4.)