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A193787 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.

%I A193787 #5 Mar 30 2012 18:57:38
%S A193787 1,1,1,1,1,2,1,2,1,4,1,3,3,1,8,1,4,6,4,1,16,1,5,10,10,5,1,32,1,6,15,
%T A193787 20,15,6,1,64,1,7,21,35,35,21,7,1,128,1,8,28,56,70,56,28,8,1,256,1,9,
%U A193787 36,84,126,126,84,36,9,1,512,1,10,45,120,210,252,210,120,45,10,1
%N A193787 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.
%C A193787 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.  A193787 is the mirror (obtained by reversing rows) of A193554.
%e A193787 First six rows:
%e A193787 1
%e A193787 1....1
%e A193787 1....1....2
%e A193787 1....2....1....4
%e A193787 1....3....3....1...8
%e A193787 1....4....6....4...1...16
%e A193787 (viz., Pascal's triangle with row sum at end of each row)
%t A193787 z = 12; a = 1; b = 1;
%t A193787 p[n_, x_] := (a*x + b)^n
%t A193787 q[n_, x_] := 1 + x^n ; q[n_, 0] := q[n, x] /. x -> 0;
%t A193787 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193787 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193787 g[n_] := CoefficientList[w[n, x], {x}]
%t A193787 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193787 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193787 *)
%t A193787 TableForm[Table[g[n], {n, -1, z}]]
%t A193787 Flatten[Table[g[n], {n, -1, z}]]  (* A193554 *)
%Y A193787 Cf. A193722, A193554.
%K A193787 nonn,tabl
%O A193787 0,6
%A A193787 _Clark Kimberling_, Aug 05 2011