A208606 - OEIS

%I #9 Sep 03 2021 13:58:02

%S 1,1,1,1,4,1,1,9,4,1,1,16,12,7,1,1,25,30,26,7,1,1,36,65,73,32,10,1,1,

%T 49,126,175,113,52,10,1,1,64,224,378,332,199,61,13,1,1,81,372,756,852,

%U 634,277,87,13,1,1,100,585,1422,1974,1779,1024,421,99,16,1,1,121

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208607; see the Formula section.

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

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

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

%e First five rows:

%e 1

%e 1...1

%e 1...4....1

%e 1...9....4....1

%e 1...16...12...7...1

%e First five polynomials u(n,x):

%e 1

%e 1 + x

%e 1 + 4x + 1x^2

%e 1 + 9x + 4x^2 + x^3

%e 1 + 16x + 12x^2 + 7x^3 + x^4

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

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

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

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]  (* A208606 *)

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

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

%t TableForm[cv]

%t Flatten[%]  (* A208607 *)

%Y Cf. A208607.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Feb 29 2012

%E a(68) corrected by _Georg Fischer_, Sep 03 2021