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A125172 Triangle T(n,k) with partial column sums of the even-indexed Fibonacci numbers.

%I A125172 #16 Mar 30 2024 23:09:30
%S A125172 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 A125172 30,8,1,987,609,370,211,103,38,9,1,2584,1596,979,581,314,141,47,10,1
%N A125172 Triangle T(n,k) with partial column sums of the even-indexed Fibonacci numbers.
%C A125172 "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 A125172 Mirror of the fission triangle A193667, as in the Mathematica program below. - _Clark Kimberling_, Aug 11 2011
%F A125172 T(n,1) = A001906(n).
%F A125172 T(n,k) = T(n-1,k-1) + T(n-1,k), k > 1.
%F A125172 From _R. J. Mathar_, Sep 06 2011: (Start)
%F A125172 T(n,k) = A125171(n,k), i.e., A125171 without column k=0.
%F A125172 Conjecture: T(n,k) = T(n,k-1) - A121460(n+1,k). (End)
%e A125172 First few rows of the triangle:
%e A125172     1;
%e A125172     3,  1;
%e A125172     8,  4,  1;
%e A125172    21, 12,  5,  1;
%e A125172    55, 33, 17,  6,  1;
%e A125172   144, 88, 50, 23,  7,  1;
%e A125172   ...
%t A125172 z = 11;
%t A125172 p[n_, x_] := (x + 1)^n;
%t A125172 q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A125172 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A125172 p1[n_, 0] := p[n, x] /. x -> 0;
%t A125172 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A125172 h[n_] := CoefficientList[d[n, x], {x}]
%t A125172 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A125172 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193667 *)
%t A125172 TableForm[Table[h[n], {n, 0, z}]]
%t A125172 Flatten[Table[h[n], {n, -1, z}]]  (* this sequence *)
%t A125172 (* _Clark Kimberling_, Aug 11 2011 *)
%Y A125172 Cf. A105693 (row sums), A125171, A193667.
%K A125172 nonn,tabl,easy
%O A125172 1,2
%A A125172 _Gary W. Adamson_, Nov 22 2006