A193955 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)=sum{((k+1)^2)*x^(n-k) : 0<=k<=n}.

1, 1, 4, 1, 5, 13, 2, 9, 23, 45, 3, 14, 36, 71, 120, 5, 23, 59, 116, 196, 300, 8, 37, 95, 187, 316, 484, 692, 13, 60, 154, 303, 512, 784, 1121, 1524, 21, 97, 249, 490, 828, 1268, 1813, 2465, 3225, 34, 157, 403, 793, 1340, 2052, 2934, 3989, 5219, 6625, 55

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..55.

Example

First six rows:
1
1...4
1...5....13
2...9....23...45
3...14...36...71....120
5...23...59...116...196...300

Mathematica

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[((k + 1)^2)*x^(n - k), {k, 0, n}] ;
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}]]  (* A193955 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193956 *)

Crossrefs

Cf. A193722, A193956.

Sequence in context: A120868 A100279 A132379 * A130746 A342925 A125078

Adjacent sequences: A193952 A193953 A193954 * A193956 A193957 A193958

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 10 2011

Status

approved