A125172 Triangle T(n,k) with partial column sums of the even-indexed Fibonacci numbers.
1, 3, 1, 8, 4, 1, 21, 12, 5, 1, 55, 33, 17, 6, 1, 144, 88, 50, 23, 7, 1, 377, 232, 138, 73, 30, 8, 1, 987, 609, 370, 211, 103, 38, 9, 1, 2584, 1596, 979, 581, 314, 141, 47, 10, 1
Offset: 1
Examples
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;
...
Programs
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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 *)
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)
Comments