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A005490 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)
4
1, 4, 13, 44, 163, 666, 2985, 14550, 76497, 430746, 2582447, 16403028, 109918745, 774289168, 5715471605, 44087879136, 354521950931, 2965359744446, 25749723493073, 231719153184018, 2157494726318233, 20753996174222510, 205985762120971167, 2106795754056142536 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Old name was: From expansion of falling factorials.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
FORMULA
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
From Alois P. Heinz, Jan 05 2022: (Start)
a(n) = Sum_{k=0..n-1} A108087(n-k,k).
a(n) = A000110(n) + Sum_{k=1..n-1} A259691(n,k)/k.
a(n) = A347420(n) - 1.
a(n) mod 2 = n mod 2 = A000035(n). (End)
EXAMPLE
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.
MAPLE
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):
seq(a(n), n=1..24); # Alois P. Heinz, Jan 05 2022
MATHEMATICA
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}];
Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Apr 24 2022, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A149430 A124463 A081197 * A174299 A351028 A285188
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Jul 01 2016
New name from Alois P. Heinz, Jan 07 2022
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)