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A347006 E.g.f.: Product_{k>=1} (1 + exp(x) * x^k / k!).

%I A347006 #15 Jul 17 2025 11:58:27
%S A347006 1,1,3,10,43,206,1044,5909,38371,272314,1995208,14869889,115433344,
%T A347006 965259881,8773348601,84608514095,837220780691,8334354200226,
%U A347006 83498917650084,855936118936073,9180736840445788,104439240481045949,1253608634906635901
%N A347006 E.g.f.: Product_{k>=1} (1 + exp(x) * x^k / k!).
%C A347006 a(n) is the number of words of length n over an infinite alphabet such that for any letter k appearing within a word, exactly k occurrences of that letter are marked. - _John Tyler Rascoe_, Jul 16 2025
%H A347006 Alois P. Heinz, <a href="/A347006/b347006.txt">Table of n, a(n) for n = 0..612</a> (first 101 terms from John Tyler Rascoe)
%F A347006 E.g.f.: exp( Sum_{k>=1} ( Sum_{d|k} (-1)^(d+1) * exp(d*x) / (d * ((k/d)!)^d) ) * x^k ).
%F A347006 E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} binomial(j,k) * x^j / j!).
%e A347006 a(3) = 10 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2#), (2#,1#,2#), (2#,2#,1#), (2#,2#,2), (2#,2,2#), (2,2#,2#), (3#,3#,3#). - _John Tyler Rascoe_, Jul 16 2025
%p A347006 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A347006       b(n, i-1)+add(b(n-j, min(n-j, i-1))/i!/(j-i)!, j=i..n)))
%p A347006     end:
%p A347006 a:= n-> n!*b(n$2):
%p A347006 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jul 17 2025
%t A347006 nmax = 22; CoefficientList[Series[Product[(1 + Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%o A347006 (PARI)
%o A347006 C_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + exp(x)*x^k/k!)))} \\ _John Tyler Rascoe_, Jul 16 2025
%Y A347006 Cf. A007837, A265952, A305547, A347005.
%K A347006 nonn,easy
%O A347006 0,3
%A A347006 _Ilya Gutkovskiy_, Aug 10 2021