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A051340
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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,...
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31
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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
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OFFSET
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0,3
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COMMENTS
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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
A member of the accumulation chain:
(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)
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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MATHEMATICA
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(* 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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