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).

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, 232, 377, 1, 9, 38, 103, 211, 370, 609, 987, 1, 10, 47, 141, 314, 581, 979, 1596, 2584, 1, 11, 57, 188, 455, 895, 1560, 2575, 4180, 6765, 1, 12, 68, 245, 643, 1350, 2455

Offset

0,3

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). A193667 is the mirror of A125172.

Links

Table of n, a(n) for n=0..61.

Example

First six rows:
1
1...3
1...4...8
1...5...12...21
1...6...17...33...55
1...7...23...50...88...144

Mathematica

z = 11;
p[n_, x_] := (x + 1)^n;
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}]]  (* A193667 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A125172 *)

Crossrefs

Cf. A193842, A125172.

Sequence in context: A005371 A210739 A193605 * A205878 A329130 A057049

Adjacent sequences: A193664 A193665 A193666 * A193668 A193669 A193670

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 11 2011

Status

approved