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A209767
Triangle of coefficients of polynomials u(n,x) jointly generated with A209768; see the Formula section.
3
1, 1, 2, 2, 6, 5, 3, 12, 20, 12, 4, 21, 52, 63, 29, 5, 33, 109, 199, 187, 70, 6, 48, 200, 490, 700, 536, 169, 7, 66, 334, 1032, 1988, 2322, 1498, 408, 8, 87, 520, 1948, 4742, 7488, 7378, 4109, 985, 9, 111, 767, 3388, 10004, 19992, 26664, 22685, 11109
OFFSET
1,3
COMMENTS
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+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...2
2...6....5
3...12...20...12
4...21...52...63...29
First three polynomials u(n,x): 1, 1 + 2x, 2 + 6x + 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 + 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[%] (* A209767 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209768 *)
CROSSREFS
Sequence in context: A064766 A019749 A209773 * A122070 A181661 A144160
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 15 2012
STATUS
approved