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 A193959 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{((k+1)^2)*x^(n-k) : 0<=k<=n} and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers) . 2

%I #5 Mar 30 2012 18:57:39

%S 1,1,4,5,9,9,13,23,36,16,25,45,71,116,25,41,75,120,196,316,36,61,113,

%T 183,300,484,784,49,85,159,260,428,692,1121,1813,64,113,213,351,580,

%U 940,1524,2465,3989,81,145,275,456,756,1228,1993,3225,5219,8444

%N Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{((k+1)^2)*x^(n-k) : 0<=k<=n} and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers) .

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

%e First six rows:

%e 1

%e 1....1

%e 4....5....9

%e 9....13...23...36

%e 16...25...45...71....116

%e 25...41...75...120...196...316

%t z = 12;

%t p[n_, x_] := Sum[((k + 1)^2)*x^(n - k), {k, 0, n}]

%t q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193959 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193960 *)

%Y Cf. A193722, A193960 .

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 10 2011

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Last modified August 15 15:45 EDT 2024. Contains 375173 sequences. (Running on oeis4.)