A193816 - OEIS

%I #18 Jan 26 2020 01:01:29

%S 1,1,2,1,5,6,1,7,17,14,1,9,31,49,30,1,11,49,111,129,62,1,13,71,209,

%T 351,321,126,1,15,97,351,769,1023,769,254,1,17,127,545,1471,2561,2815,

%U 1793,510,1,19,161,799,2561,5503,7937,7423,4097,1022,1,21,199,1121

%N 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) = (x+2)^n.

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

%C Triangle T(n,k), read by rows, given by (1,0,-1,1,0,0,0,0,0,0,0,...) DELTA (2,1,-2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011

%F T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k-1) - 2*T(n-2,k-2), T(0,0)=1, T(1,0)=1, T(1,1)=2, T(2,0)=1, T(2,1)=5, T(2,2)=6, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Dec 15 2013

%F G.f.: (1-x*y+x^2*y+2*x^2*y^2)/((-1+x+2*x*y)*(x*y-1)). - _R. J. Mathar_, Aug 12 2015

%e First six rows:

%e   1;

%e   1,  2;

%e   1,  5,  6;

%e   1,  7, 17,  14;

%e   1,  9, 31,  49,  30;

%e   1, 11, 49, 111, 129, 62;

%t z = 10; c = 1; d = 2;

%t p[0, x_] := 1

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

%t q[n_, x_] := (c*x + d)^n

%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}]]  (* A193816 *)

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

%t Flatten[Table[g[n], {n, -1, z}]]   (* A193817 *)

%Y Cf. A084938, A193722, A193817.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 06 2011