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A209574
Triangle of coefficients of polynomials v(n,x) jointly generated with A209573; see the Formula section.
3
1, 2, 3, 3, 7, 5, 4, 14, 20, 7, 5, 24, 52, 45, 9, 6, 37, 110, 155, 86, 11, 7, 53, 203, 403, 389, 147, 13, 8, 72, 340, 882, 1240, 856, 232, 15, 9, 94, 530, 1712, 3204, 3322, 1702, 345, 17, 10, 119, 782, 3040, 7170, 10088, 7962, 3125, 490, 19, 11, 147, 1105
OFFSET
1,2
COMMENTS
Combinatorial limit of row n satisfies linear recurrence
r(n)=4*r(n-1)-r(n-2) with r(1)=1 and r(2)=3. For a
discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+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
2...3
3...7....5
4...14...20...7
5...24...52...45...9
First three polynomials v(n,x): 1, 2 + 3x , 3 + 7x + 5x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 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[%] (* A209573 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209574 *)
CROSSREFS
Sequence in context: A358182 A208525 A342878 * A079387 A141061 A256447
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 11 2012
STATUS
approved