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A209169 Triangle of coefficients of polynomials v(n,x) jointly generated with A209168; see the Formula section. 3
1, 2, 3, 3, 7, 7, 5, 16, 23, 17, 8, 33, 65, 70, 41, 13, 65, 159, 233, 204, 99, 21, 124, 362, 654, 776, 577, 239, 34, 231, 782, 1676, 2447, 2461, 1597, 577, 55, 423, 1627, 4018, 6937, 8586, 7534, 4348, 1393, 89, 764, 3289, 9179, 18202, 26597, 28750 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Column 1: Fibonacci numbers (A000045).
For a discussion and guide to related arrays, see A208510.
LINKS
FORMULA
u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
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) = 1, T(2,0) = 2, T(2,1) = 3, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, 11 2012
EXAMPLE
First five rows:
1
2...3
3...7....7
5...16...23...17
8...33...65...70...41
MATHEMATICA
First three polynomials v(n, x): 1, 2 + 3x, 3 + 7x + 7x^2.
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] + 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[%] (* A209168 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209169 *)
CROSSREFS
Sequence in context: A208920 A210234 A209768 * A222294 A181850 A335050
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 08 2012
STATUS
approved

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Last modified April 24 00:30 EDT 2024. Contains 371917 sequences. (Running on oeis4.)