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 (2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x*y)/(1-2*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A000079(n).
T(n, 1) = A130129(n-1).
T(n, n) = A133494(n).
T(n, n-1) = A199923(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)
EXAMPLE
First six rows:
1;
2, 1;
4, 8, 3;
8, 28, 30, 9;
16, 80, 144, 108, 27;
32, 208, 528, 648, 378, 81;
MATHEMATICA
(* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; 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}]] (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)
PROG
(Magma)
function T(n, k) // T = A193730
if k lt 0 or k gt n then return 0;
elif n lt 2 then return n-k+1;
else return 2*T(n-1, k) + 3*T(n-1, k-1);
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023
(SageMath)
def T(n, k): # T = A193730
if (k<0 or k>n): return 0
elif (n<2): return n-k+1
else: return 2*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved