A194009 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

2, 3, 5, 4, 7, 13, 5, 9, 17, 28, 6, 11, 21, 35, 58, 7, 13, 25, 42, 70, 114, 8, 15, 29, 49, 82, 134, 218, 9, 17, 33, 56, 94, 154, 251, 407, 10, 19, 37, 63, 106, 174, 284, 461, 747, 11, 21, 41, 70, 118, 194, 317, 515, 835, 1352, 12, 23, 45, 77, 130, 214, 350, 569

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:
2
3...5
4...7....13
5...9....17...28
6...11...21...35...58
7...13...25...42...70...114
		

Crossrefs

Cf. A193842, A194010.

Programs

Mathematica

z = 11;
p[n_, x_] := x*p[n - 1, x] + n + 1; p[0, n_] := 1;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
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}]]  (* A194009 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194010 *)