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A209765
Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section.
3
1, 1, 2, 1, 5, 5, 1, 5, 15, 12, 1, 5, 21, 45, 29, 1, 5, 21, 77, 129, 70, 1, 5, 21, 89, 265, 361, 169, 1, 5, 21, 89, 353, 865, 991, 408, 1, 5, 21, 89, 377, 1325, 2717, 2681, 985, 1, 5, 21, 89, 377, 1549, 4733, 8281, 7169, 2378, 1, 5, 21, 89, 377, 1597, 6125
OFFSET
1,3
COMMENTS
Limiting row: F(2+3k), where F=A000045 (Fibonacci numbers)
Coefficient of x^n in u(n,x): 1,2,5,12,.... A000129(n)
Row sums: 1,3,11,33,101,303,911,2733,..... A081250
Alternating row sums: 1,-1,1,-1,1,-1,,..... A033999
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=2x*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...2
1...5...5
1...5...15...12
1...5...21...45...29
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^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] + 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[%] (* A209765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209766 *)
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}] (* A042963 signed *)
CROSSREFS
Sequence in context: A005605 A300661 A145882 * A209759 A111785 A304462
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved