A210800 Triangle of coefficients of polynomials v(n,x) jointly generated with A210799; see the Formula section.

1, 1, 2, 5, 4, 3, 5, 14, 9, 5, 17, 28, 36, 19, 8, 17, 70, 88, 83, 38, 13, 53, 136, 251, 245, 181, 73, 21, 53, 298, 557, 746, 613, 379, 137, 34, 161, 568, 1376, 1930, 2030, 1439, 769, 252, 55, 161, 1162, 2888, 5026, 5818, 5139, 3221, 1524, 457, 89, 485

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

Row n starts a term of A048473 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Alternating row sums: 1,2,3,4,5,6,7,...

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

Examples

			First five rows:
1
1....2
5....4....3
5....14...9....5
17...28...36...19...8
First three polynomials v(n,x): 1, 1 + 2x, 5 + 4x + 3x^2
		

Crossrefs

Cf. A210799, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 1; h = 2; p = -1; f = 0;
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[%]   (* A210799 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210800 *)

Formula

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

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

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

T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2) + a(k) with a(0) = 2, a(1) = 1, a(k) = 0 if k>1, T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k >n. - Philippe Deléham, Mar 31 2012