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