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A208508 Triangle of coefficients of polynomials u(n,x) jointly generated with A208509; see the Formula section.

%I A208508 #10 Nov 07 2015 13:57:40
%S A208508 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,
%T A208508 112,10,1,81,336,294,54,1,1,100,540,672,210,12,1,121,825,1386,660,77,
%U A208508 1,1,144,1210,2640,1782,352,14,1,169,1716,4719,4290,1287,104,1,1
%N A208508 Triangle of coefficients of polynomials u(n,x) jointly generated with A208509; see the Formula section.
%C A208508 col 1:  A000012
%C A208508 col 2:  A000290 (squares)
%C A208508 col 3:  A002415
%C A208508 col 4:  A040977
%C A208508 col 5:  A054334
%C A208508 row sums, u(n,1): A083329
%F A208508 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208508 v(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A208508 where u(1,x)=1, v(1,x)=1.
%e A208508 First five rows:
%e A208508 1
%e A208508 1...1
%e A208508 1...4
%e A208508 1...9....1
%e A208508 1...16...6
%e A208508 First five polynomials u(n,x):
%e A208508 1
%e A208508 1 + x
%e A208508 1 + 4x
%e A208508 1 + 9x + x^2
%e A208508 1 + 16x + 6x^2
%t A208508 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208508 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208508 v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A208508 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208508 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208508 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208508 TableForm[cu]
%t A208508 Flatten[%]    (* A208508 *)
%t A208508 Table[Expand[v[n, x]], {n, 1, z}]
%t A208508 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208508 TableForm[cv]
%t A208508 Flatten[%]    (* A208509 *)
%Y A208508 Cf. A208509.
%K A208508 nonn,tabf
%O A208508 1,5
%A A208508 _Clark Kimberling_, Feb 27 2012