A193816 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.

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, 351, 321, 126, 1, 15, 97, 351, 769, 1023, 769, 254, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 510, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1022, 1, 21, 199, 1121

Offset

0,3

Comments

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

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

Links

Table of n, a(n) for n=0..58.

Formula

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

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

Example

First six rows:
  1;
  1,  2;
  1,  5,  6;
  1,  7, 17,  14;
  1,  9, 31,  49,  30;
  1, 11, 49, 111, 129, 62;

Mathematica

z = 10; c = 1; d = 2;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193816 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193817 *)

Crossrefs

Cf. A084938, A193722, A193817.

Sequence in context: A209108 A269019 A184234 * A335136 A193722 A193635

Adjacent sequences: A193813 A193814 A193815 * A193817 A193818 A193819

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 06 2011

Status

approved