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A210552
Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.
3
1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73
OFFSET
1,3
COMMENTS
Let T(n,k) denote the term in row n, column k.
T(n,n): A000045 (Fibonacci numbers)
T(n,n-1): A010049 (second-order Fibonacci numbers)
T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...
T(n,2): 2,3,4,5,6,7,8,9,10,11,...
T(n,3): 3,5,7,9,11,13,15,17,19,...
T(n,4): A052905
Row sums: A000225
Alternating row sums: A094024 (signed)
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)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...2
1...3...3
1...4...5...5
1...5...7...10...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.
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_] := x*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[%] (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)
CROSSREFS
Sequence in context: A210489 A344821 A125175 * A193376 A185095 A177888
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 22 2012
STATUS
approved