login
Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.
3

%I #22 Nov 07 2023 19:48:18

%S 1,1,1,1,2,1,1,3,4,2,1,4,8,8,3,1,5,13,19,15,5,1,6,19,36,42,28,8,1,7,

%T 26,60,91,89,51,13,1,8,34,92,170,216,182,92,21,1,9,43,133,288,446,489,

%U 363,164,34,1,10,53,184,455,826,1105,1068,709,290,55,1,11,64,246,682,1414,2219,2619,2266,1362,509,89

%N Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

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

%H G. C. Greubel, <a href="/A193736/b193736.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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

%F From _G. C. Greubel_, Oct 24 2023: (Start)

%F T(n, k) = A193737(n, n-k).

%F T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).

%F T(n, n-1) = A029907(n).

%F Sum_{k=0..n} T(n, k) = A052542(n).

%F Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].

%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)

%e First six rows:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 4, 2;

%e 1, 4, 8, 8, 3;

%e 1, 5, 13, 19, 15, 5;

%t (* First program *)

%t p[0, x_] := 1

%t p[n_, x_] := Fibonacci[n + 1, x] /; n > 0

%t q[n_, x_] := (x + 1)^n

%t t[n_, k_] := Coefficient[p[n, x], x^(n - k)];

%t t[n_, n_] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n - k + 1, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193736 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193737 *)

%t (* Second program *)

%t 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]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* _G. C. Greubel_, Oct 24 2023 *)

%o (Magma)

%o function T(n,k) // T = A193736

%o if k lt 0 or n lt 0 then return 0;

%o elif n lt 3 then return Binomial(n,k);

%o else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);

%o end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 24 2023

%o (SageMath)

%o def T(n,k): # T = A193736

%o if (n<3): return binomial(n,k)

%o else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k-2)

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 24 2023

%Y Cf. A000007, A005314 (diagonal sums), A052542 (row sums), A077962.

%Y Cf. A029907, A193722, A193737, A324969.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 04 2011