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A209148
Triangle of coefficients of polynomials u(n,x) jointly generated with A209149; see the Formula section.
3
1, 2, 1, 5, 5, 1, 11, 16, 7, 1, 23, 44, 30, 9, 1, 47, 112, 104, 48, 11, 1, 95, 272, 320, 200, 70, 13, 1, 191, 640, 912, 720, 340, 96, 15, 1, 383, 1472, 2464, 2352, 1400, 532, 126, 17, 1, 767, 3328, 6400, 7168, 5152, 2464, 784, 160, 19, 1, 1535, 7424
OFFSET
1,2
COMMENTS
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = u(n-1,x) + (x+1)*v(n-1,x)+1,
where u(1,x) = 1, v(1,x) = 1.
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k)-T(n-2,k-1), T(1,0)=T(2,1)=T(3,2)=1, T(2,0)=2, T(3,0)=T(3,1)=5, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Jan 17 2014
EXAMPLE
First five rows:
1
2....1
5....5....1
11...16...7....1
23...44...30...9...1
First three polynomials v(n,x): 1, 2 + x, 5 + 5x + x^2.
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] + (x + 1)*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[%] (* A209148 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209149 *)
CROSSREFS
Sequence in context: A204111 A079502 A209164 * A126124 A123971 A060920
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 07 2012
EXTENSIONS
Example corrected by Philippe Deléham, Jan 17 2014
STATUS
approved