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%I #16 Mar 30 2024 23:09:30
%S 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,
%T 30,8,1,987,609,370,211,103,38,9,1,2584,1596,979,581,314,141,47,10,1
%N Triangle T(n,k) with partial column sums of the even-indexed Fibonacci numbers.
%C "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
%C Mirror of the fission triangle A193667, as in the Mathematica program below. - _Clark Kimberling_, Aug 11 2011
%F T(n,1) = A001906(n).
%F T(n,k) = T(n-1,k-1) + T(n-1,k), k > 1.
%F From _R. J. Mathar_, Sep 06 2011: (Start)
%F T(n,k) = A125171(n,k), i.e., A125171 without column k=0.
%F Conjecture: T(n,k) = T(n,k-1) - A121460(n+1,k). (End)
%e First few rows of the triangle:
%e 1;
%e 3, 1;
%e 8, 4, 1;
%e 21, 12, 5, 1;
%e 55, 33, 17, 6, 1;
%e 144, 88, 50, 23, 7, 1;
%e ...
%t z = 11;
%t p[n_, x_] := (x + 1)^n;
%t q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t p1[n_, k_] := Coefficient[p[n, x], x^k];
%t p1[n_, 0] := p[n, x] /. x -> 0;
%t d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t h[n_] := CoefficientList[d[n, x], {x}]
%t TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A193667 *)
%t TableForm[Table[h[n], {n, 0, z}]]
%t Flatten[Table[h[n], {n, -1, z}]] (* this sequence *)
%t (* _Clark Kimberling_, Aug 11 2011 *)
%Y Cf. A105693 (row sums), A125171, A193667.
%K nonn,tabl,easy
%O 1,2
%A _Gary W. Adamson_, Nov 22 2006
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