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A208514 Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section.

%I A208514 #8 Dec 20 2021 06:50:23
%S A208514 1,1,1,1,2,2,1,3,4,3,1,4,6,7,5,1,5,8,12,13,8,1,6,10,18,24,23,13,1,7,
%T A208514 12,25,38,46,41,21,1,8,14,33,55,78,88,72,34,1,9,16,42,75,120,158,165,
%U A208514 126,55,1,10,18,52,98,173,255,313,307,219,89,1,11,20,63,124,238
%N A208514 Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section.
%C A208514 u(n,n) = Fibonacci(n), A000045
%C A208514 u(n+1,n) = A208354(n)
%C A208514 col 1:  A000012
%C A208514 col 2:  A000027
%C A208514 col 3:  A005843
%C A208514 col 4:  A055998
%C A208514 col 5:  A140090
%F A208514 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208514 v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A208514 where u(1,x)=1, v(1,x)=1.
%e A208514 First five rows:
%e A208514 1
%e A208514 1...1
%e A208514 1...2...2
%e A208514 1...3...4...3
%e A208514 1...4...6...7...5
%e A208514 First five polynomials u(n,x):
%e A208514 1
%e A208514 1 + x
%e A208514 1 + 2x + 2x^2
%e A208514 1 + 3x + 4x^2 + 3x^3
%e A208514 1 + 4x + 6x^2 + 7x^3 + 5x^4
%t A208514 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208514 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208514 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A208514 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208514 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208514 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208514 TableForm[cu]
%t A208514 Flatten[%]    (* A208514 *)
%t A208514 Table[Expand[v[n, x]], {n, 1, z}]
%t A208514 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208514 TableForm[cv]
%t A208514 Flatten[%]    (* A208515 *)
%Y A208514 Cf. A208515.
%K A208514 nonn,tabl
%O A208514 1,5
%A A208514 _Clark Kimberling_, Feb 28 2012