A193997 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+1)^n.

1, 2, 3, 3, 8, 6, 5, 18, 23, 11, 8, 37, 66, 55, 19, 13, 73, 167, 196, 120, 32, 21, 139, 388, 587, 511, 246, 53, 34, 259, 853, 1578, 1777, 1225, 484, 87, 55, 474, 1799, 3933, 5428, 4857, 2765, 924, 142, 89, 856, 3678, 9275, 15147, 16642, 12333, 5969

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....3
3....8....6
5....18...23....11
8....37...66....55....19
13...73...167...196...120...32
		

Crossrefs

Cf. A193842, A193998.

Programs

Mathematica

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := (x + 1)^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}]]  (* A193997 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193998 *)