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 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,... 31
 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, 1, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 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 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,1)=n and T(n,k)=1 if k>1, for n>=1. - 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 A051340 (i.e., the array of which A051340 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 *) CROSSREFS Cf. A144112, A141419, A185874, A185875, A185876. Sequence in context: A080209 A127949 A167407 * A216764 A165430 A334215 Adjacent sequences:  A051337 A051338 A051339 * A051341 A051342 A051343 KEYWORD easy,nice,nonn,tabl AUTHOR EXTENSIONS Edited by M. F. Hasler, Aug 15 2015 STATUS approved

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Last modified July 28 14:19 EDT 2021. Contains 346335 sequences. (Running on oeis4.)