A185045 Triangle of coefficients of polynomials u(n,x) jointly generated with A208659; see the Formula section.

1, 1, 2, 1, 6, 4, 1, 10, 16, 8, 1, 14, 36, 40, 16, 1, 18, 64, 112, 96, 32, 1, 22, 100, 240, 320, 224, 64, 1, 26, 144, 440, 800, 864, 512, 128, 1, 30, 196, 728, 1680, 2464, 2240, 1152, 256, 1, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512, 1, 38, 324

Offset

1,3

Comments

Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,-1,...

For a discussion and guide to related arrays, see A208510.

Links

Table of n, a(n) for n=1..58.

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),

v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1), T(1,0) = T(2,0) = T(3,0) = 1, T(2,1) = 2, T(3,1) = 6, T(3,2) = 4, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 19 2012

Example

First five rows:
1
1...2
1...6...4
1...10...16...8
1...14...36...40...16
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 4x^2
1 + 10x + 16x^2 + 8x^3
1 + 14x + 36x^2 + 40x^3 + 16x^4

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_] := u[n - 1, x] + 2 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[%]    (* A185045 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208659 *)

Crossrefs

Cf. A208659, A208510.

Sequence in context: A349573 A378413 A208923 * A208913 A208911 A208761

Adjacent sequences: A185042 A185043 A185044 * A185046 A185047 A185048

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2012

Status

approved