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%I #11 Jun 23 2018 02:33:15
%S 1,1,1,3,2,1,5,6,3,1,9,13,10,4,1,15,28,26,15,5,1,25,56,64,45,21,6,1,
%T 41,109,146,124,71,28,7,1,67,206,319,315,216,105,36,8,1,109,382,671,
%U 758,602,349,148,45,9,1,177,697,1372,1744,1576,1056,533,201,55,10
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209578; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x) = x*u(n-1,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.
%F The coefficients in the triangle seem to be T(n,m) = sum(k=0,n-m,2 * binomial(m+k, m)*binomial(k, n-k-m) - sum(i=0, n-m-k, binomial(m+k-1,k)*binomial(k,n-m-i-k))) by using the PARI syntax. - _Thomas Baruchel_, Jun 03 2018
%e First five rows:
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 5, 6, 3, 1;
%e 9, 13, 10, 4, 1;
%e First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, 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[%] (* A209577 *)
%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[%] (* A209578 *)
%Y Cf. A209578, A208510.
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_, Mar 11 2012
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