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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.

%I A347420 #48 Apr 05 2023 16:56:39
%S A347420 1,2,5,14,45,164,667,2986,14551,76498,430747,2582448,16403029,
%T A347420 109918746,774289169,5715471606,44087879137,354521950932,
%U A347420 2965359744447,25749723493074,231719153184019,2157494726318234,20753996174222511,205985762120971168,2106795754056142537
%N 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.
%H A347420 Alois P. Heinz, <a href="/A347420/b347420.txt">Table of n, a(n) for n = 0..575</a>
%H A347420 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A347420 a(n) = Sum_{k=0..n} A108087(n-k,k).
%F A347420 a(n) = 1 + A005490(n).
%F A347420 a(n) = A000110(n) + Sum_{k=1..n} k * A259691(n-1,k).
%F A347420 a(n) = Sum_{k=1..n} (k+1) * A259691(n-1,k).
%F A347420 a(n) = A000110(n) + A350589(n).
%F A347420 a(n) mod 2 = A059841(n).
%e A347420 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 A347420 b:= proc(n, m) option remember;
%p A347420      `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
%p A347420     end:
%p A347420 a:= n-> add(b(i, n-i), i=0..n):
%p A347420 seq(a(n), n=0..25);
%t A347420 b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]];
%t A347420 a[n_] := Sum[b[i, n - i], {i, 0, n}];
%t A347420 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 11 2022, after _Alois P. Heinz_ *)
%Y A347420 Antidiagonal sums of A108087.
%Y A347420 Cf. A000110, A005490, A059841, A259691, A350589, A361380.
%K A347420 nonn
%O A347420 0,2
%A A347420 _Alois P. Heinz_, Jan 05 2022