A210866 Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.

1, 1, 1, 1, 3, 1, 1, 6, 6, 2, 1, 10, 21, 14, 3, 1, 15, 55, 65, 31, 5, 1, 21, 120, 235, 187, 65, 8, 1, 28, 231, 700, 867, 503, 134, 13, 1, 36, 406, 1792, 3332, 2914, 1279, 268, 21, 1, 45, 666, 4074, 10955, 13882, 9084, 3122, 527, 34, 1, 55, 1035, 8430, 31563

Offset

1,5

Comments

Row n starts with 1, followed by the n-th triangular number, and ends with the n-th Fibonacci number.

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

Links

Table of n, a(n) for n=1..60.

Formula

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

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

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

Example

First five rows:
1
1...1
1...3....1
1...6....6....2
1...10...21...14...3
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - 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[%]   (* A210866 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210867 *)

Crossrefs

Cf. A210867, A208510.

Sequence in context: A172427 A143362 A182823 * A245474 A338369 A339231

Adjacent sequences: A210863 A210864 A210865 * A210867 A210868 A210869

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 29 2012

Status

approved