A193953 Triangular array: the fusion 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, n>=0.

1, 1, 2, 1, 3, 5, 2, 5, 9, 13, 3, 8, 14, 21, 28, 5, 13, 23, 34, 46, 58, 8, 21, 37, 55, 74, 94, 114, 13, 34, 60, 89, 120, 152, 185, 218, 21, 55, 97, 144, 194, 246, 299, 353, 407, 34, 89, 157, 233, 314, 398, 484, 571, 659, 747, 55, 144, 254, 377, 508, 644, 783

Offset

0,3

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Links

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

Example

First six rows:
1
1...2
1...3....5
2...5....9....13
3...8....14...21...28
5...13...23...34...46...58

Mathematica

z = 12;
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, x_] := 1
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193953 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193954 *)

Crossrefs

Cf. A193722, A193954.

Sequence in context: A210880 A210867 A019588 * A201377 A368070 A322942

Adjacent sequences: A193950 A193951 A193952 * A193954 A193955 A193956

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 10 2011

Status

approved