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A210744
Triangle of coefficients of polynomials v(n,x) jointly generated with A210743; see the Formula section.
3
1, 3, 1, 6, 6, 3, 11, 18, 18, 7, 19, 45, 63, 49, 17, 32, 100, 182, 200, 133, 41, 53, 208, 464, 658, 613, 356, 99, 87, 413, 1094, 1886, 2244, 1823, 944, 239, 142, 794, 2437, 4940, 7093, 7325, 5302, 2483, 577, 231, 1490, 5206, 12113, 20311, 25220
OFFSET
1,2
COMMENTS
Row n starts with -2+F(n+3), where F=A000045 (Fibonacci numbers) and ends with A001333(n-1).
Alternating row sums: 1,2,3,4,5,6,7,8,...
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
3....1
6....6....3
11...18...18...7
19...45...63...49...17
First three polynomials v(n,x): 1, 3 + x, 6 + 6x +3x^2
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, 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[%] (* A210743 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210744 *)
CROSSREFS
Sequence in context: A116412 A089511 A246257 * A242729 A112692 A291217
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 24 2012
STATUS
approved