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)

1, 4, 13, 44, 163, 666, 2985, 14550, 76497, 430746, 2582447, 16403028, 109918745, 774289168, 5715471605, 44087879136, 354521950931, 2965359744446, 25749723493073, 231719153184018, 2157494726318233, 20753996174222510, 205985762120971167, 2106795754056142536

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

Table of n, a(n) for n=1..24.

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

Cf. A000035, A000110, A108087, A259691, A347420.

Sequence in context: A149430 A124463 A081197 * A174299 A351028 A285188

Adjacent sequences: A005487 A005488 A005489 * A005491 A005492 A005493

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Sean A. Irvine, Jul 01 2016

New name from Alois P. Heinz, Jan 07 2022

Status

approved