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A209753
Triangle of coefficients of polynomials u(n,x) jointly generated with A209754; see the Formula section.
3
1, 1, 2, 3, 5, 3, 5, 14, 12, 4, 9, 30, 40, 23, 5, 15, 63, 107, 93, 39, 6, 25, 124, 264, 300, 190, 61, 7, 41, 238, 604, 858, 722, 354, 90, 8, 67, 445, 1319, 2242, 2364, 1559, 615, 127, 9, 109, 818, 2772, 5500, 6966, 5783, 3101, 1011, 173, 10, 177, 1482
OFFSET
1,3
COMMENTS
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
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)=u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...2
3...5....3
5...14...12...4
9...30...40...23...5
First three polynomials u(n,x): 1, 1 + 2x, 3 + 5x + 3x^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_] := u[n - 1, x] + (x + 1)*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[%] (* A209753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209754 *)
CROSSREFS
Sequence in context: A336746 A209769 A114230 * A185191 A103781 A095244
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved