A209571 Triangle of coefficients of polynomials u(n,x) jointly generated with A209572; see the Formula section.

1, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 4, 15, 16, 1, 1, 4, 15, 44, 25, 1, 1, 4, 15, 56, 105, 36, 1, 1, 4, 15, 56, 185, 216, 49, 1, 1, 4, 15, 56, 209, 524, 399, 64, 1, 1, 4, 15, 56, 209, 732, 1295, 680, 81, 1, 1, 4, 15, 56, 209, 780, 2303, 2864, 1089, 100, 1, 1, 4, 15, 56

Offset

1,5

Comments

Penultimate number in row n is (n-1)^2, for n>1.

Combinatorial limit of row n satisfies linear recurrence

r(n)=4*r(n-1)-r(n-2) with r(1)=1 and r(2)=4. For a

discussion and guide to related arrays, see A208510.

Links

Table of n, a(n) for n=1..70.

Formula

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

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

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

Example

First five rows:
1
1...1
1...4....1
1...4....9....1
1...4....15...16...1
First three polynomials v(n,x): 1, 1 + x, 1 + 4x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 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[%]   (* A209571 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209572 *)

Crossrefs

Cf. A209572, A208510.

Sequence in context: A301626 A080061 A246595 * A269845 A124258 A001638

Adjacent sequences: A209568 A209569 A209570 * A209572 A209573 A209574

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 11 2012

Status

approved