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A193740 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x) in A193738.

%I A193740 #5 Mar 30 2012 18:57:38
%S A193740 1,1,1,1,3,3,1,4,9,9,1,4,10,19,19,1,4,10,20,34,34,1,4,10,20,35,55,55,
%T A193740 1,4,10,20,35,56,83,83,1,4,10,20,35,56,84,119,119,1,4,10,20,35,56,84,
%U A193740 120,164,164,1,4,10,20,35,56,84,120,165,219,219,1,4,10,20,35,56
%N A193740 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x) in A193738.
%C A193740 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%e A193740 First six rows:
%e A193740 1
%e A193740 1....1
%e A193740 1....3....3
%e A193740 1....4....9....9
%e A193740 1....4....10....19...19
%e A193740 1....4....10....20...34...34
%t A193740 z = 12;
%t A193740 p[0, x_] := 1
%t A193740 p[n_, x_] := n + Sum[(k + 1) x^(n - k), {k, 0, n - 1}]
%t A193740 q[n_, x_] := p[n, x]
%t A193740 t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
%t A193740 t[n_, n_] := p[n, x] /. x -> 0;
%t A193740 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193740 g[n_] := CoefficientList[w[n, x], {x}]
%t A193740 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193740 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193740 *)
%t A193740 TableForm[Table[g[n], {n, -1, z}]]
%t A193740 Flatten[Table[g[n], {n, -1, z}]]   (* A193741 *)
%Y A193740 Cf. A193722, A193741.
%K A193740 nonn,tabl
%O A193740 0,5
%A A193740 _Clark Kimberling_, Aug 04 2011