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A338817
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Matrix inverse of triangle A176270, read by rows.
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1
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1, -1, 1, -1, 0, 1, -3, 1, 1, 1, -12, 4, 5, 2, 1, -60, 20, 25, 11, 3, 1, -360, 120, 150, 66, 19, 4, 1, -2520, 840, 1050, 462, 133, 29, 5, 1, -20160, 6720, 8400, 3696, 1064, 232, 41, 6, 1, -181440, 60480, 75600, 33264, 9576, 2088, 369, 55, 7, 1, -1814400, 604800, 756000, 332640, 95760, 20880, 3690, 550, 71, 8, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,7
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LINKS
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FORMULA
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T(n,n) = 1 for n >= 0; T(n,n-1) = n - 2 for n > 0; T(n,n-2) = n^2 - 3*n + 1 for n > 1; T(n,k) = (k^2 + k - 1) * n! / (k+2)! for 0 <= k <= n-2.
T(n,k) = n * T(n-1,k) for 0 <= k < n-2.
T(n,k) = T(k+2,k) * n! / (k+2)! for 0 <= k <= n-2.
Row sums are A000007(n) for n >= 0.
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EXAMPLE
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The triangle T(n,k) for 0 <= k <= n starts:
n\k : 0 1 2 3 4 5 6 7 8 9
=============================================================
0 : 1
1 : -1 1
2 : -1 0 1
3 : -3 1 1 1
4 : -12 4 5 2 1
5 : -60 20 25 11 3 1
6 : -360 120 150 66 19 4 1
7 : -2520 840 1050 462 133 29 5 1
8 : -20160 6720 8400 3696 1064 232 41 6 1
9 : -181440 60480 75600 33264 9576 2088 369 55 7 1
etc.
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PROG
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(PARI) for(n=0, 10, for(k=0, n, if(k==n, print(" 1"), if(k==n-1, print1(n-2, ", "), print1((k^2+k-1)*n!/(k+2)!, ", ")))))
(PARI) 1/matrix(10, 10, n, k, n--; k--; if (n>=k, 1 + k*(k-n))) \\ Michel Marcus, Nov 11 2020
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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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