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A193734
Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=(x+2)^n.
4
1, 1, 2, 1, 6, 8, 1, 10, 32, 32, 1, 14, 72, 160, 128, 1, 18, 128, 448, 768, 512, 1, 22, 200, 960, 2560, 3584, 2048, 1, 26, 288, 1760, 6400, 13824, 16384, 8192, 1, 30, 392, 2912, 13440, 39424, 71680, 73728, 32768, 1, 34, 512, 4480, 25088, 93184, 229376, 360448, 327680, 131072
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 (1,0,0,0,0,0,0,0,...) DELTA (2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011
FORMULA
T(n,k) = 4*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1 - 2*x*y)/(1 - x - 4*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 19 2023: (Start)
T(n, n) = A081294(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A026581(n-1) + (1/2)*[n=0]. (End)
EXAMPLE
First six rows:
1;
1, 2;
1, 6, 8;
1, 10, 32, 32;
1, 14, 72, 160, 128;
1, 18, 128, 448, 768, 512;
MATHEMATICA
(* First program *)
z = 8; a = 2; b = 1; c = 1; d = 2;
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}]] (* A193734 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193735 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, T[n-1, k] +4*T[n-1, k-1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//TableForm (* G. C. Greubel, Nov 19 2023 *)
PROG
(Magma)
function T(n, k) // T = A193734
if k lt 0 or k gt n then return 0;
elif n lt 2 then return k+1;
else return T(n-1, k) + 4*T(n-1, k-1);
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2023
(SageMath)
def T(n, k): # T = A193734
if (k<0 or k>n): return 0
elif (n<2): return k+1
else: return T(n-1, k) +4*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 19 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved