%I #48 Apr 05 2023 16:56:39

%S 1,2,5,14,45,164,667,2986,14551,76498,430747,2582448,16403029,

%T 109918746,774289169,5715471606,44087879137,354521950932,

%U 2965359744447,25749723493074,231719153184019,2157494726318234,20753996174222511,205985762120971168,2106795754056142537

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

%H Alois P. Heinz, <a href="/A347420/b347420.txt">Table of n, a(n) for n = 0..575</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

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

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

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

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

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

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

%e 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'.

%p b:= proc(n, m) option remember;

%p `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))

%p end:

%p a:= n-> add(b(i, n-i), i=0..n):

%p seq(a(n), n=0..25);

%t b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]];

%t a[n_] := Sum[b[i, n - i], {i, 0, n}];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 11 2022, after _Alois P. Heinz_ *)

%Y Antidiagonal sums of A108087.

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

%K nonn

%O 0,2

%A _Alois P. Heinz_, Jan 05 2022