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A193736 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.

%I A193736 #22 Nov 07 2023 19:48:18
%S A193736 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 A193736 26,60,91,89,51,13,1,8,34,92,170,216,182,92,21,1,9,43,133,288,446,489,
%U A193736 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 A193736 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 A193736 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%H A193736 G. C. Greubel, <a href="/A193736/b193736.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193736 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 A193736 From _G. C. Greubel_, Oct 24 2023: (Start)
%F A193736 T(n, k) = A193737(n, n-k).
%F A193736 T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).
%F A193736 T(n, n-1) = A029907(n).
%F A193736 Sum_{k=0..n} T(n, k) = A052542(n).
%F A193736 Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
%F A193736 Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].
%F A193736 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)
%e A193736 First six rows:
%e A193736   1;
%e A193736   1,  1;
%e A193736   1,  2,  1;
%e A193736   1,  3,  4,  2;
%e A193736   1,  4,  8,  8,  3;
%e A193736   1,  5, 13, 19, 15,  5;
%t A193736 (* First program *)
%t A193736 p[0, x_] := 1
%t A193736 p[n_, x_] := Fibonacci[n + 1, x] /; n > 0
%t A193736 q[n_, x_] := (x + 1)^n
%t A193736 t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
%t A193736 t[n_, n_] := p[n, x] /. x -> 0;
%t A193736 w[n_, x_] := Sum[t[n, k]*q[n - k + 1, x], {k, 0, n}]; w[-1, x_] := 1
%t A193736 g[n_] := CoefficientList[w[n, x], {x}]
%t A193736 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193736 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193736 *)
%t A193736 TableForm[Table[g[n], {n, -1, z}]]
%t A193736 Flatten[Table[g[n], {n, -1, z}]]          (* A193737 *)
%t A193736 (* Second program *)
%t A193736 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 A193736 Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* _G. C. Greubel_, Oct 24 2023 *)
%o A193736 (Magma)
%o A193736 function T(n,k) // T = A193736
%o A193736   if k lt 0 or n lt 0 then return 0;
%o A193736   elif n lt 3 then return Binomial(n,k);
%o A193736   else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);
%o A193736   end if;
%o A193736 end function;
%o A193736 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 24 2023
%o A193736 (SageMath)
%o A193736 def T(n,k): # T = A193736
%o A193736     if (n<3): return binomial(n,k)
%o A193736     else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k-2)
%o A193736 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 24 2023
%Y A193736 Cf. A000007, A005314 (diagonal sums), A052542 (row sums), A077962.
%Y A193736 Cf. A029907, A193722, A193737, A324969.
%K A193736 nonn,tabl
%O A193736 0,5
%A A193736 _Clark Kimberling_, Aug 04 2011