A194007 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers) and q(n,x)=x*q(n-1,x)+n+1, with q(0,x)=1.

1, 2, 5, 3, 8, 14, 5, 13, 23, 34, 8, 21, 37, 55, 74, 13, 34, 60, 89, 120, 152, 21, 55, 97, 144, 194, 246, 299, 34, 89, 157, 233, 314, 398, 484, 571, 55, 144, 254, 377, 508, 644, 783, 924, 1066, 89, 233, 411, 610, 822, 1042, 1267, 1495, 1725, 1956, 144, 377

Offset: 0

Author

Clark Kimberling, Aug 11 2011

Keywords

• nonn
• tabl

Comments

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).

Examples

			First six rows:
1
2....5
3....8....14
5....13...23...34
8....21...37...55...74
13...34...60...89...120...152
		

Crossrefs

Cf. A193842, A194008.

Programs

Mathematica

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := x*q[n - 1, x] + n + 1; q[0, n_] := 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194007 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194008 *)