A051340 A simple 2-dimensional array, read by antidiagonals: T[i,j] = 1 for j>0, T[i,0] = i+1; i,j = 0,1,2,3,...

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset

0,3

Comments

Warning: contributions from Kimberling refer to an alternate version indexed by 1 instead of 0. Other contributors (Adamson in A125026/A130301/A130295) refer to this considering the upper right triangle set to zero, T[i,j]=0 for j>i. - M. F. Hasler, Aug 15 2015

From Clark Kimberling, Feb 05 2011: (Start)

A member of the accumulation chain:

... < A051340 < A141419 < A185874 < A185875 < A185876 < ...

(See A144112 for the definition of accumulation array.)

In the m-th accumulation array of A051340,

row_1 = C(m,1) and column_1 = C(1,m+1), for m>=0. (End)

Links

G. C. Greubel, Antidiagonals n = 0..100, flattened

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

A. V. Mikhalev and A. A. Nechaev, Linear recurring sequences over modules, Acta Applic. Math., 42 (1996), 161-202.

Formula

For n>0, a(n(n+3)/2)=n+1, and if k is not of the form n*(n+3)/2, then a(k)=1. - Benoit Cloitre, Oct 31 2002, corrected by M. F. Hasler, Aug 15 2015

T(n,0) = n+1 and T(n,k) = 1 if k >= 0, for n >= 0. - Clark Kimberling, Feb 05 2011

Example

Northwest corner:
  1...1...1...1...1...1...1
  2...1...1...1...1...1...1
  3...1...1...1...1...1...1
  4...1...1...1...1...1...1
  5...1...1...1...1...1...1
  6...1...1...1...1...1...1
The Mathematica code shows that the weight array of this array (i.e., the array of which this array is the accumulation array), has northwest corner
  1....0...0...0...0...0...0
  1...-1...0...0...0...0...0
  1...-1...0...0...0...0...0
  1...-1...0...0...0...0...0
  1...-1...0...0...0...0...0. - Clark Kimberling, Feb 05 2011

Maple

A051340 := proc(n, k) if k=0 then n+1; else 1; end if; end proc: # R. J. Mathar, Jul 16 2015

Mathematica

(* This program generates A051340, then its accumulation array A141419, then its weight array described under Example. *)
f[n_, 0]:=0; f[0, k_]:=0;  (* needed for the weight array *)
f[n_, 1]:=n; f[n_, k_]:=1/; k>1;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]] (* A051340 *)
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
s[n_, k_]:=Sum[f[i, j], {i, 1, n}, {j, 1, k}]; (* accumulation array of {f(n, k)} *)
TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]] (* A141419 *)
Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
w[m_, n_]:=f[m, n]+f[m-1, n-1]-f[m, n-1]-f[m-1, n]/; Or[m>0, n>0];
TableForm[Table[w[n, k], {n, 1, 10}, {k, 1, 15}]] (* weight array *)
Table[w[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten (* Clark Kimberling, Feb 05 2011 *)
f[n_] := Join[ Table[1, {n - 1}], {n}]; Array[ f, 14] // Flatten (* Robert G. Wilson v, Mar 04 2012 *)

Prog

(Magma) [k eq n select n+1 else 1: k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 18 2023
(SageMath)
def A051340(n, k): return n+1 if (k==n) else 1
flatten([[A051340(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 18 2023

Crossrefs

Cf. A144112, A141419, A185874, A185875, A185876.

Sequence in context: A080209 A347171 A127949 * A167407 A216764 A165430

Adjacent sequences: A051337 A051338 A051339 * A051341 A051342 A051343

Keyword

easy,nice,nonn,tabl

Author

N. J. A. Sloane

Extensions

Edited by M. F. Hasler, Aug 15 2015

Status

approved