login
A209764
Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.
3
1, 2, 2, 3, 4, 4, 4, 8, 14, 8, 5, 14, 32, 34, 16, 6, 22, 62, 96, 86, 32, 7, 32, 108, 218, 280, 202, 64, 8, 44, 174, 432, 718, 760, 470, 128, 9, 58, 264, 778, 1584, 2194, 1992, 1066, 256, 10, 74, 382, 1304, 3142, 5360, 6382, 5048, 2390, 512, 11, 92, 532
OFFSET
1,2
COMMENTS
Row n begins with n and ends with 2^(n-1).
Row sums: 1,4,11,34,101,304,911,2734,... A060925.
Alternating row sums: 1,0,3,2,5,4,7,6,... A060925.
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
2...2
3...4....4
4...8....14...8
5...14...32...34...16
First three polynomials v(n,x): 1, 2 + 2x , 3 + 4x + 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: A359897 A209698 A141525 * A071475 A358104 A343228
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved