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A005490
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Number of partitions of [n] where the first k elements are marked (0 <= k <= n-1) and at least k blocks contain their own index.
(Formerly M3467)
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4
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1, 4, 13, 44, 163, 666, 2985, 14550, 76497, 430746, 2582447, 16403028, 109918745, 774289168, 5715471605, 44087879136, 354521950931, 2965359744446, 25749723493073, 231719153184018, 2157494726318233, 20753996174222510, 205985762120971167, 2106795754056142536
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OFFSET
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1,2
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COMMENTS
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Old name was: From expansion of falling factorials.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n} b(n, i) where b(n, 1) = n and b(n+1, i+1) = (n-i) * b(n, i) + b(n+1, i) [From Whitehead]. - Sean A. Irvine, Jul 01 2016
a(n) = Sum_{k=0..n-1} A108087(n-k,k).
a(n) mod 2 = n mod 2 = A000035(n). (End)
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EXAMPLE
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a(3) = 13 = 5 + 5 + 3: 123, 12|3, 13|2, 1|23, 1|2|3, 1'23, 1'2|3, 1'3|2, 1'|23, 1'|2|3, 1'3|2', 1'|2'3, 1'|2'|3.
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MAPLE
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b:= proc(n, m) option remember;
`if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
end:
a:= n-> add(b(n-k, k), k=0..n-1):
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MATHEMATICA
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b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]];
a[n_] := Sum[b[n - k, k], {k, 0, n - 1}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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