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A209144
Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.
5
1, 3, 6, 1, 12, 5, 24, 16, 1, 48, 44, 7, 96, 112, 30, 1, 192, 272, 104, 9, 384, 640, 320, 48, 1, 768, 1472, 912, 200, 11, 1536, 3328, 2464, 720, 70, 1, 3072, 7424, 6400, 2352, 340, 13, 6144, 16384, 16128, 7168, 1400, 96, 1, 12288, 35840, 39680, 20736
OFFSET
1,2
COMMENTS
Alternating row sums: 1,3,5,7,9,11,13,15,17,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (3,-1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012
FORMULA
u(n,x) = u(n-1,x) + (x+1)*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.
From Philippe Deléham, Mar 07 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x)/(1-2*x-y*x^2).
Sum_{k=0..n} T(n,k)*x^k = A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. (End)
EXAMPLE
First five rows:
1;
3;
6, 1;
12, 5;
24, 16, 1;
First three polynomials v(n,x): 1, 3, 6 + x.
(3,-1, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, ...) begins:
1;
3, 0;
6, 1, 0;
12, 5, 0, 0;
24, 16, 1, 0, 0;
48, 44, 7, 0, 0, 0;
96, 112, 30, 1, 0, 0, 0;
192, 272, 104, 9, 0, 0, 0, 0;
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_] := 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[%] (* A209143 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209144 *)
CROSSREFS
Sequence in context: A210039 A026250 A210033 * A130724 A120229 A266151
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Mar 06 2012
STATUS
approved