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A210551
Triangle of coefficients of polynomials v(n,x) jointly generated with A172431; see the Formula section.
2
1, 3, 1, 5, 6, 1, 7, 15, 10, 1, 9, 28, 35, 15, 1, 11, 45, 84, 70, 21, 1, 13, 66, 165, 210, 126, 28, 1, 15, 91, 286, 495, 462, 210, 36, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 21, 190, 969, 3060, 6188
OFFSET
1,2
COMMENTS
Row sums: -1+odd-indexed Fibonacci numbers
Alternating row sums: 1,2,0,1,2,0,1,2,0,...
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1
where u(1,x)=1, v(1,x)=1.
EXAMPLE
From Paul Weisenhorn, May 17 2020 : (Start)
First five rows of v(n.x):
1
3 1
5 6 1
7 15 10 1
9 28 35 15 1
First three polynomials v(n,x): 1, 3 + x, 5 + 6x + x^2. (End)
From Paul Weisenhorn, May 14 2020: (Start)
First five rows of u(n,x):
1
1 2
1 4 3
1 6 10 4
1 8 21 20 5
First three polynomials u(n,x): 1, 1 + 2x, 1 + 4x + 3x^2. (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%] (* A172431 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210551 *)
CROSSREFS
Sequence in context: A089028 A209758 A134083 * A113445 A108283 A208904
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 22 2012
STATUS
approved