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A209569
Triangle of coefficients of polynomials u(n,x) jointly generated with A209570; see the Formula section.
3
1, 1, 1, 2, 3, 1, 3, 6, 5, 1, 4, 11, 14, 7, 1, 5, 18, 31, 26, 9, 1, 6, 27, 60, 71, 42, 11, 1, 7, 38, 105, 162, 139, 62, 13, 1, 8, 51, 170, 327, 372, 243, 86, 15, 1, 9, 66, 259, 602, 861, 754, 391, 114, 17, 1, 10, 83, 376, 1031, 1790, 1987, 1388, 591, 146, 19, 1
OFFSET
1,4
COMMENTS
For n>1, row n begins with n and ends with 2.
Alternating row sums: 1,0,0,1,1,0,0,1,1,0,0,1,1,...
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)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...1
2...3....1
3...6....5....1
4...11...14...7...1
First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^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] + 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[%] (* A209569 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209570 *)
CROSSREFS
Sequence in context: A256193 A101912 A208522 * A368158 A176850 A208516
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 10 2012
STATUS
approved