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A278644 Number of partitions of n into parts of sorts {1, 2, ... }.

%I A278644 #24 Sep 26 2023 03:08:59
%S A278644 1,1,4,17,95,649,5423,53345,604570,7744990,110596370,1740967790,
%T A278644 29943077149,558541778035,11229820022013,242071441524480,
%U A278644 5568954194762675,136181762611151941,3527284819779421843,96465042641948254298,2777679881076121497601
%N A278644 Number of partitions of n into parts of sorts {1, 2, ... }.
%C A278644 Parts are unordered, sorts are ordered, all sorts up to the highest have to be present.
%C A278644 a(n) mod 2 = A040051(n).
%H A278644 Alois P. Heinz, <a href="/A278644/b278644.txt">Table of n, a(n) for n = 0..424</a>
%F A278644 a(n) = Sum_{k=0..n} A255970(n,k).
%F A278644 a(n) = Sum_{k=0..n} A008284(n,k) * A000670(k). - _Ludovic Schwob_, Sep 25 2023
%F A278644 a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Sep 26 2023
%e A278644 a(3) = 17: 1a1a1a, 2a1a, 1a, 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a (in this example the sorts are labeled a, b, c).
%p A278644 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A278644       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A278644     end:
%p A278644 a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p A278644 seq(a(n), n=0..25);
%t A278644 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 06 2017, translated from Maple *)
%Y A278644 Row sums of A255970.
%Y A278644 Cf. A000670, A008284, A040051, A258466.
%K A278644 nonn
%O A278644 0,3
%A A278644 _Alois P. Heinz_, Nov 24 2016