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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. 4

%I

%S 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 243,1,13,73,233,473,665,729,729,1,15,99,379,939,1611,2059,2187,2187,

%U 1,17,129,577,1697,3489,5281,6305,6561,6561,1,19,163,835,2851

%N 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 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

%F From _Peter Bala_, Jul 16 2013: (Start)

%F T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.

%F 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 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 First six rows:

%e 1

%e 1....1

%e 1....3....3

%e 1....5....9....9

%e 1....7....19...27...27

%e 1....9....33...65...81...81

%t z = 10; a = 2; b = 1;

%t p[n_, x_] := (a*x + b)^n

%t q[0, x_] := 1

%t q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193823 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193824 *)

%Y Cf. A193722, A193804. A119258, A193860.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 06 2011

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Last modified February 20 09:15 EST 2019. Contains 320325 sequences. (Running on oeis4.)