A208508 - OEIS

1, 1, 1, 1, 4, 1, 9, 1, 1, 16, 6, 1, 25, 20, 1, 1, 36, 50, 8, 1, 49, 105, 35, 1, 1, 64, 196, 112, 10, 1, 81, 336, 294, 54, 1, 1, 100, 540, 672, 210, 12, 1, 121, 825, 1386, 660, 77, 1, 1, 144, 1210, 2640, 1782, 352, 14, 1, 169, 1716, 4719, 4290, 1287, 104, 1, 1

FORMULA

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

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

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

EXAMPLE

First five rows:

1...1

1...4

1...9....1

1...16...6

First five polynomials u(n,x):

1 + x

1 + 4x

1 + 9x + x^2

1 + 16x + 6x^2

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

v[n_, x_] := 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]

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]