OFFSET
1,2
COMMENTS
"Partial column sums" means the 1st column consists of the even-indexed Fibonacci numbers, the 2nd column shows the partial sums of the first column, the 3rd column the partial sums of the 2nd, etc. - R. J. Mathar, Sep 06 2011
Mirror of the fission triangle A193667, as in the Mathematica program below. - Clark Kimberling, Aug 11 2011
FORMULA
T(n,1) = A001906(n).
T(n,k) = T(n-1,k-1) + T(n-1,k), k > 1.
From R. J. Mathar, Sep 06 2011: (Start)
Conjecture: T(n,k) = T(n,k-1) - A121460(n+1,k). (End)
EXAMPLE
First few rows of the triangle:
1;
3, 1;
8, 4, 1;
21, 12, 5, 1;
55, 33, 17, 6, 1;
144, 88, 50, 23, 7, 1;
...
MATHEMATICA
z = 11;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A193667 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]] (* this sequence *)
(* Clark Kimberling, Aug 11 2011 *)
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Nov 22 2006
STATUS
approved