A121460 -id:A121460 - cp's OEIS frontend

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

Gary W. Adamson, Nov 22 2006

"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

			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;
  ...

Cf. A105693 (row sums), A125171, A193667.

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 *)

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)
T(n,k) = A125171(n,k), i.e., A125171 without column k=0.
Conjecture: T(n,k) = T(n,k-1) - A121460(n+1,k). (End)

A328083 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and height k (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 11, 14, 7, 1, 1, 19, 35, 24, 9, 1, 1, 32, 80, 72, 36, 11, 1, 1, 53, 173, 196, 123, 50, 13, 1

Offset: 1

N. J. A. Sloane, Oct 16 2019

			Triangle begins:
1,
1,1,
1,3,1,
1,6,5,1,
1,11,14,7,1,
1,19,35,24,9,1,
1,32,80,72,36,11,1,
1,53,173,196,123,50,13,1,
...

Rigoberto Flórez, Leandro Junes, José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3089.

Cf. A121460.

cp's OEIS Frontend

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

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