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A220417
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Table T(n,k) = k^n - n^k, n, k > 0, read by descending antidiagonals.
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4
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0, 1, -1, 2, 0, -2, 3, 1, -1, -3, 4, 0, 0, 0, -4, 5, -7, -17, 17, 7, -5, 6, -28, -118, 0, 118, 28, -6, 7, -79, -513, -399, 399, 513, 79, -7, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, 9, -431, -6049, -13983, -7849, 7849, 13983, 6049, 431, -9, 10, -924, -18954, -61440, -61318, 0, 61318, 61440, 18954, 924, -10
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = ((t*t + 3*t + 4)/2 - n)^(n - t*(t + 1)/2) - (n - t*(t + 1)/2)^((t*t + 3*t + 4)/2 - n) where t = floor((-1 + sqrt(8*n - 7))/2).
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EXAMPLE
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The table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
0 1 2 3 4 5 ...
-1 0 1 0 -7 -28 ...
-2 -1 0 -17 -118 -513 ...
-3 0 17 0 -399 -2800 ...
-4 7 118 399 0 -7849 ...
-5 28 513 2800 7849 0 ...
...
The start of the sequence as a triangular array, read by rows (i.e., descending antidiagonals of T(n,k)), is as follows:
0;
1, -1;
2, 0, -2;
3, 1, -1, -3;
4, 0, 0, 0, -4;
5, -7, -17, 17, 7, -5;
6, -28, -118, 0, 118, 28, -6;
...
In the above triangle, row number m contains m numbers: m^1 - 1^m, (m-1)^2 - 2^(m-1), ..., 1^m - m^1.
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PROG
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(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
m=((t*t+3*t+4)/2-n)**(n-t*(t+1)/2)-(n-t*(t+1)/2)**((t*t+3*t+4)/2-n)
(PARI) matrix(9, 9, n, k, k^n - n^k) \\ Michel Marcus, Oct 04 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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