cp's OEIS Frontend

A193667 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

%I A193667 #7 Dec 26 2023 09:55:25
%S A193667 1,1,3,1,4,8,1,5,12,21,1,6,17,33,55,1,7,23,50,88,144,1,8,30,73,138,
%T A193667 232,377,1,9,38,103,211,370,609,987,1,10,47,141,314,581,979,1596,2584,
%U A193667 1,11,57,188,455,895,1560,2575,4180,6765,1,12,68,245,643,1350,2455
%N A193667 Triangular array:  the fission of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).
%C A193667 See A193842 for the definition of the fission of P by Q, where P and Q are sequences of polynomials or triangular arrays (of coefficients of polynomials).  A193667 is the mirror of A125172.
%e A193667 First six rows:
%e A193667 1
%e A193667 1...3
%e A193667 1...4...8
%e A193667 1...5...12...21
%e A193667 1...6...17...33...55
%e A193667 1...7...23...50...88...144
%t A193667 z = 11;
%t A193667 p[n_, x_] := (x + 1)^n;
%t A193667 q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A193667 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193667 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193667 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193667 h[n_] := CoefficientList[d[n, x], {x}]
%t A193667 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193667 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193667 *)
%t A193667 TableForm[Table[h[n], {n, 0, z}]]
%t A193667 Flatten[Table[h[n], {n, -1, z}]]  (* A125172 *)
%Y A193667 Cf. A193842, A125172.
%K A193667 nonn,tabl
%O A193667 0,3
%A A193667 _Clark Kimberling_, Aug 11 2011