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A193951
Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{(k+1)(n+1)*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
1, 1, 1, 4, 6, 10, 9, 15, 27, 42, 16, 28, 52, 84, 136, 25, 45, 85, 140, 230, 370, 36, 66, 126, 210, 348, 564, 912, 49, 91, 175, 294, 490, 798, 1295, 2093, 64, 120, 232, 392, 656, 1072, 1744, 2824, 4568, 81, 153, 297, 504, 846, 1386, 2259, 3663, 5931, 9594
OFFSET
0,4
COMMENTS
See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
EXAMPLE
First six rows:
1
1....1
4....6....10
9....15...27...42
16...28...52...84....136
25...45...85...140...230...370
MATHEMATICA
z = 12;
p[n_, x_] := Sum[(k + 1) (n + 1)*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[Fibonacci[k + 1]*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}]] (* A193951 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193952 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 10 2011
STATUS
approved