OFFSET
0,5
COMMENTS
See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2). - Philippe Deléham, Feb 13 2020
From G. C. Greubel, Oct 24 2023: (Start)
T(n, k) = A193737(n, n-k).
T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).
T(n, n-1) = A029907(n).
Sum_{k=0..n} T(n, k) = A052542(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)
EXAMPLE
First six rows:
1;
1, 1;
1, 2, 1;
1, 3, 4, 2;
1, 4, 8, 8, 3;
1, 5, 13, 19, 15, 5;
MATHEMATICA
(* First program *)
p[0, x_] := 1
p[n_, x_] := Fibonacci[n + 1, x] /; n > 0
q[n_, x_] := (x + 1)^n
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n - k + 1, 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}]] (* A193736 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193737 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1, k] +T[n-1, k-1] +T[n -2, k-2]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//TableForm (* G. C. Greubel, Oct 24 2023 *)
PROG
(Magma)
function T(n, k) // T = A193736
if k lt 0 or n lt 0 then return 0;
elif n lt 3 then return Binomial(n, k);
else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023
(SageMath)
def T(n, k): # T = A193736
if (n<3): return binomial(n, k)
else: return T(n-1, k) +T(n-1, k-1) +T(n-2, k-2)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved