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A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

%I A193823 #12 Jul 17 2013 01:14:03
%S A193823 1,1,1,1,3,3,1,5,9,9,1,7,19,27,27,1,9,33,65,81,81,1,11,51,131,211,243,
%T A193823 243,1,13,73,233,473,665,729,729,1,15,99,379,939,1611,2059,2187,2187,
%U A193823 1,17,129,577,1697,3489,5281,6305,6561,6561,1,19,163,835,2851
%N A193823 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.
%C A193823 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%F A193823 From _Peter Bala_, Jul 16 2013: (Start)
%F A193823 T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.
%F A193823 O.g.f.: (1 - 2*x*t)^2/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + x)*t + (1 + 3*x + 3*x^2)*t^2 + .... Cf. A193860.
%F A193823 For n >= 1, the n-th row polynomial R(n,x) = 1/(x-1)*( 3^(n-1)*x^(n+1) - (2*x + 1)^(n-1) ). (End)
%e A193823 First six rows:
%e A193823 1
%e A193823 1....1
%e A193823 1....3....3
%e A193823 1....5....9....9
%e A193823 1....7....19...27...27
%e A193823 1....9....33...65...81...81
%t A193823 z = 10; a = 2; b = 1;
%t A193823 p[n_, x_] := (a*x + b)^n
%t A193823 q[0, x_] := 1
%t A193823 q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
%t A193823 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193823 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193823 g[n_] := CoefficientList[w[n, x], {x}]
%t A193823 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193823 Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193823 *)
%t A193823 TableForm[Table[g[n], {n, -1, z}]]
%t A193823 Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)
%Y A193823 Cf. A193722, A193804. A119258, A193860.
%K A193823 nonn,tabl
%O A193823 0,5
%A A193823 _Clark Kimberling_, Aug 06 2011