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A193818 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = x^n + x^(n-1) + ... + x+1 and q(n,x)=(2x+1)^n.

%I A193818 #22 Jan 22 2020 16:23:30
%S A193818 1,2,1,4,6,2,8,16,12,3,16,40,40,20,4,32,96,120,80,30,5,64,224,336,280,
%T A193818 140,42,6,128,512,896,896,560,224,56,7,256,1152,2304,2688,2016,1008,
%U A193818 336,72,8,512,2560,5760,7680,6720,4032,1680,480,90,9,1024,5632
%N A193818 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x) = x^n + x^(n-1) + ... + x+1 and q(n,x)=(2x+1)^n.
%C A193818 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193818 Triangle T(n,k), read by rows, given by (2,0,-2,2,0,0,0,0,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011
%F A193818 T(n,k) = A193815(n,k)*2(n-k).
%F A193818 T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2), T(0,0)=T(1,1)=1, T(1,0)=2, T(2,0)=4, T(2,1)=6, T(2,2)=2, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Dec 15 2013
%F A193818 G.f.: (1-x*y+2*x^2*y+x^2*y^2)/((-1+2*x+x*y)*(x*y-1)). - _R. J. Mathar_, Aug 11 2015
%e A193818 First six rows:
%e A193818    1;
%e A193818    2,  1;
%e A193818    4,  6,   2;
%e A193818    8, 16,  12,  3;
%e A193818   16, 40,  40, 20,  4;
%e A193818   32, 96, 120, 80, 30, 5;
%t A193818 z = 10; c = 2; d = 1;
%t A193818 p[0, x_] := 1
%t A193818 p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
%t A193818 q[n_, x_] := (c*x + d)^n
%t A193818 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193818 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193818 g[n_] := CoefficientList[w[n, x], {x}]
%t A193818 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193818 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193818 *)
%t A193818 TableForm[Table[g[n], {n, -1, z}]]
%t A193818 Flatten[Table[g[n], {n, -1, z}]]   (* A193819 *)
%Y A193818 Cf. A084938, A193815, A193722, A193819.
%K A193818 nonn,tabl
%O A193818 0,2
%A A193818 _Clark Kimberling_, Aug 06 2011