A209577 Triangle of coefficients of polynomials u(n,x) jointly generated with A209578; see the Formula section.

1, 1, 1, 3, 2, 1, 5, 6, 3, 1, 9, 13, 10, 4, 1, 15, 28, 26, 15, 5, 1, 25, 56, 64, 45, 21, 6, 1, 41, 109, 146, 124, 71, 28, 7, 1, 67, 206, 319, 315, 216, 105, 36, 8, 1, 109, 382, 671, 758, 602, 349, 148, 45, 9, 1, 177, 697, 1372, 1744, 1576, 1056, 533, 201, 55, 10

Offset: 1

Author

Clark Kimberling, Mar 11 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
  1;
  1,  1;
  3,  2,  1;
  5,  6,  3,  1;
  9, 13, 10,  4,  1;
First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.
		

Crossrefs

Cf. A209578, A208510.

Programs

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_] := (x + 1)*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[%]   (* A209577 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209578 *)

Formula

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

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

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

The coefficients in the triangle seem to be T(n,m) = sum(k=0,n-m,2 * binomial(m+k, m)*binomial(k, n-k-m) - sum(i=0, n-m-k, binomial(m+k-1,k)*binomial(k,n-m-i-k))) by using the PARI syntax. - Thomas Baruchel, Jun 03 2018