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A209761
Triangle of coefficients of polynomials u(n,x) jointly generated with A209762; see the Formula section.
3
1, 1, 2, 2, 5, 4, 3, 10, 14, 8, 4, 17, 34, 36, 16, 5, 26, 68, 104, 88, 32, 6, 37, 120, 240, 296, 208, 64, 7, 50, 194, 480, 776, 800, 480, 128, 8, 65, 294, 868, 1736, 2352, 2080, 1088, 256, 9, 82, 424, 1456, 3472, 5824, 6784, 5248, 2432, 512, 10, 101, 588
OFFSET
1,3
COMMENTS
Column 1: 1,1,2,3,4,5,6,7,...
Column 2: 1+1, 1+2^2, 1+3^2, 1+4^2,...
Last term in row n: 2^(n-1)
Alternating row sums: 1,-1,1,-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)=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
2...5....4
3...10...14...8
4...17...34...36...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^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_] := 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[%] (* A209761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209762 *)
CROSSREFS
Sequence in context: A275381 A283235 A209763 * A228526 A209745 A249620
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved