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A209422
Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section.
3
1, 3, 5, 1, 9, 2, 1, 15, 6, 2, 1, 25, 13, 7, 2, 1, 41, 28, 16, 8, 2, 1, 67, 56, 38, 19, 9, 2, 1, 109, 109, 82, 49, 22, 10, 2, 1, 177, 206, 173, 112, 61, 25, 11, 2, 1, 287, 382, 352, 252, 146, 74, 28, 12, 2, 1, 465, 697, 701, 543, 347, 184, 88, 31, 13, 2, 1, 753, 1256, 1368, 1144, 784, 459, 226, 103, 34, 14, 2
OFFSET
1,2
COMMENTS
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) = u(n-1,x) + v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
G.f.: (1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)) = 1 + 3*t + (5 + x)*t^2 + ... . - G. C. Greubel, Jan 03 2018
EXAMPLE
First five rows:
1;
3;
5, 1;
9, 2, 1;
15, 6, 2, 1;
First three polynomials v(n,x): 1, 3, 5 + x.
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_] := 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[%] (* A209421 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209422 *)
CoefficientList[CoefficientList[Series[(1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)), {t, 0, 10}], t], x]// Flatten (* G. C. Greubel, Jan 03 2018 *)
CROSSREFS
Sequence in context: A197326 A235605 A212695 * A361944 A320386 A112411
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Mar 09 2012
STATUS
approved