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A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section. 3
1, 2, 1, 2, 4, 3, 2, 6, 12, 7, 2, 8, 22, 32, 17, 2, 10, 34, 70, 86, 41, 2, 12, 48, 124, 216, 228, 99, 2, 14, 64, 196, 428, 644, 600, 239, 2, 16, 82, 288, 744, 1408, 1876, 1568, 577, 2, 18, 102, 402, 1188, 2664, 4476, 5364, 4074, 1393, 2, 20, 124, 540, 1786 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012
Row sums are powers of 3 (A000244). - Philippe Deléham, Mar 21 2012
LINKS
FORMULA
u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 21 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-2*y*x+x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
T(n,k) = T(n-1,k-1) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
EXAMPLE
First five rows:
1;
2, 1;
2, 4, 3;
2, 6, 12, 7;
2, 8, 22, 32, 17;
First three polynomials u(n,x):
1
2 + x
2 + 4x + 3x^2
From Philippe Deléham, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
2, 4, 3, 0;
2, 6, 12, 7, 0;
2, 8, 22, 32, 17, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
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[%] (* A209128 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209129 *)
CROSSREFS
Sequence in context: A099312 A117505 A331499 * A209131 A165053 A302982
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 05 2012
STATUS
approved

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Last modified May 23 07:28 EDT 2024. Contains 372760 sequences. (Running on oeis4.)