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A208956
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Triangular array read by rows. T(n,k) is the number of n-permutations that have at least k fixed points with n >= 1 and 1 <= k <= n.
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2
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1, 1, 1, 4, 1, 1, 15, 7, 1, 1, 76, 31, 11, 1, 1, 455, 191, 56, 16, 1, 1, 3186, 1331, 407, 92, 22, 1, 1, 25487, 10655, 3235, 771, 141, 29, 1, 1, 229384, 95887, 29143, 6883, 1339, 205, 37, 1, 1, 2293839, 958879, 291394, 68914, 13264, 2176, 286, 46, 1, 1
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OFFSET
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1,4
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COMMENTS
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Row sums = n!
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LINKS
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FORMULA
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E.g.f. for column k: 1/(1-x) - D(x)*Sum_{i=0..k-1} x^i/i! where D(x) is the e.g.f. for A000166.
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EXAMPLE
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Triangle begins:
1;
1, 1;
4, 1, 1;
15, 7, 1, 1;
76, 31, 11, 1, 1;
455, 191, 56, 16, 1, 1;
3186, 1331, 407, 92, 22, 1, 1;
...
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MAPLE
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b:= proc(n) b(n):= `if`(n<2, 1-n, (n-1)*(b(n-1)+b(n-2))) end:
T:= (n, k)-> add(binomial(n, i)*b(n-i), i=k..n):
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MATHEMATICA
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f[list_] := Select[list, #>0&]; Map[f, Transpose[Table[nn=10; d=Exp[-x]/(1-x); p=1/(1-x); s=Sum[x^i/i!, {i, 0, n}]; Drop[Range[0, nn]! CoefficientList[Series[p-s d, {x, 0, nn}], x], 1], {n, 0, 9}]]]//Flatten
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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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