A347420 Number of partitions of [n] where the first k elements are marked (0 <= k <= n) and at least k blocks contain their own index.

1, 2, 5, 14, 45, 164, 667, 2986, 14551, 76498, 430747, 2582448, 16403029, 109918746, 774289169, 5715471606, 44087879137, 354521950932, 2965359744447, 25749723493074, 231719153184019, 2157494726318234, 20753996174222511, 205985762120971168, 2106795754056142537

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..575

Wikipedia, Partition of a set

Formula

a(n) = Sum_{k=0..n} A108087(n-k,k).

a(n) = 1 + A005490(n).

a(n) = A000110(n) + Sum_{k=1..n} k * A259691(n-1,k).

a(n) = Sum_{k=1..n} (k+1) * A259691(n-1,k).

a(n) = A000110(n) + A350589(n).

a(n) mod 2 = A059841(n).

Example

a(3) = 14 = 5 + 5 + 3 + 1: 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, 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(i, n-i), i=0..n):
seq(a(n), n=0..25);

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[i, n - i], {i, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

Crossrefs

Antidiagonal sums of A108087.

Cf. A000110, A005490, A059841, A259691, A350589, A361380.

Sequence in context: A268004 A119429 A290615 * A174300 A059216 A259871

Adjacent sequences: A347417 A347418 A347419 * A347421 A347422 A347423

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2022

Status

approved