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.

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, 140, 42, 6, 128, 512, 896, 896, 560, 224, 56, 7, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 8, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 9, 1024, 5632

Offset

0,2

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 (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

Links

Table of n, a(n) for n=0..56.

Formula

T(n,k) = A193815(n,k)*2(n-k).

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

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

Example

First six rows:
   1;
   2,  1;
   4,  6,   2;
   8, 16,  12,  3;
  16, 40,  40, 20,  4;
  32, 96, 120, 80, 30, 5;

Mathematica

z = 10; c = 2; d = 1;
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}]]  (* A193818 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193819 *)

Crossrefs

Cf. A084938, A193815, A193722, A193819.

Sequence in context: A111932 A121456 A323286 * A127535 A285491 A257640

Adjacent sequences: A193815 A193816 A193817 * A193819 A193820 A193821

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 06 2011

Status

approved