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A209763
Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section.
3
1, 1, 2, 2, 5, 4, 3, 9, 13, 8, 4, 15, 31, 35, 16, 5, 23, 61, 97, 85, 32, 6, 33, 107, 219, 279, 203, 64, 7, 45, 173, 433, 717, 761, 469, 128, 8, 59, 263, 779, 1583, 2195, 1991, 1067, 256, 9, 75, 381, 1305, 3141, 5361, 6381, 5049, 2389, 512, 10, 93, 531
OFFSET
1,3
COMMENTS
Row n begins with n and ends with 2^(n-1).
Row sums: 1,3,11,33,101,303,911,... A081250
Alternating row sums: 1,-1,1,-1,1,.. A033999
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+(x+1)*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...2
2...5....4
3...9....13...8
4...15...31...35...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%] (* A209763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209764 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442 *)
CROSSREFS
Sequence in context: A209751 A275381 A283235 * A209761 A228526 A209745
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved