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A208755
Triangle of coefficients of polynomials u(n,x) jointly generated with A208756; see the Formula section.
4
1, 1, 2, 1, 2, 4, 1, 2, 6, 8, 1, 2, 8, 14, 16, 1, 2, 10, 20, 34, 32, 1, 2, 12, 26, 56, 78, 64, 1, 2, 14, 32, 82, 140, 178, 128, 1, 2, 16, 38, 112, 218, 352, 398, 256, 1, 2, 18, 44, 146, 312, 594, 852, 882, 512, 1, 2, 20, 50, 184, 422, 912, 1530, 2040, 1934, 1024
OFFSET
1,3
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
FORMULA
u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 04 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 1 and T(n,k) = 0 if k < 0 or if k > n. (End)
G.f.: -(1+x*y)*x*y/(-1+x*y-x^2*y+2*x^2*y^2+x). - R. J. Mathar, Aug 11 2015
EXAMPLE
First five rows:
1;
1, 2;
1, 2, 4;
1, 2, 6, 8;
1, 2, 8, 14, 16;
First five polynomials u(n,x):
1
1 + 2x
1 + 2x + 4x^2
1 + 2x + 6x^2 + 8x^3
1 + 2x + 8x^2 + 14x^3 + 16x^4
From Philippe Deléham, Mar 04 2012: (Start)
Triangle (1, 0, -1, 1, 0, 0, 0...) DELTA (0, 2, 0, -1, 0, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 2, 4, 0;
1, 2, 6, 8, 0;
1, 2, 8, 14, 16, 0;
1, 2, 10, 20, 34, 32, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 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[%] (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208756 *)
CROSSREFS
Sequence in context: A376826 A114791 A129994 * A226441 A080246 A113413
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 01 2012
STATUS
approved