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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.

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editing

_Clark Kimberling (ck6(AT)evansville.edu), _, Aug 10 2011

proposed

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editing

proposed

allocated for Clark KimberlingTriangular 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

0,3

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

First six rows:

1

1...2

1...3....5

2...5....9....13

3...8....14...21...28

5...13...23...34...46...58

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 *)

Cf. A193722, A193954.

allocated

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Aug 10 2011

approved

editing

allocated for Clark Kimberling

allocated

approved