A209140 - OEIS

A209140

Triangle of coefficients of polynomials v(n,x) jointly generated with A209139; see the Formula section.

1, 1, 3, 2, 5, 7, 3, 12, 18, 17, 5, 23, 51, 58, 41, 8, 45, 118, 189, 175, 99, 13, 84, 264, 506, 645, 507, 239, 21, 155, 558, 1268, 1950, 2085, 1428, 577, 34, 281, 1145, 2974, 5395, 6998, 6482, 3940, 1393, 55, 504, 2286, 6687, 13851, 21141, 23856

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OFFSET

1,3

COMMENTS

Column 1: Fibonacci numbers, A000045.

Alternating row sums: (-2)^(n-1).

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

LINKS

Table of n, a(n) for n=1..52.

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) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 3, T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Apr 11 2012

EXAMPLE

First five rows:

1, 3;

2, 5, 7;

3, 12, 18, 17;

5, 23, 51, 58, 41;

First three polynomials v(n,x):

1 + 3x

2 + 5x + 7x^2.

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] + 2 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[%] (* A209139 *)

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

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

TableForm[cv]

Flatten[%] (* A209140 *)

CROSSREFS

Cf. A209139, A208510.

Sequence in context: A130295 A208613 A209584 * A265903 A345420 A364885

Adjacent sequences: A209137 A209138 A209139 * A209141 A209142 A209143

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved