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A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

%I A386255 #17 Jul 18 2025 04:31:06
%S A386255 1,1,4,15,64,325,1776,11179,72640,489969,3435580,26495491,221599104,
%T A386255 1893705697,16145571820,138299146665,1241234863936,12033569772769,
%U A386255 124055067568788,1303750295285563,13577876900409280,139418829477000801,1441311794301705964,15537427948684769425
%N A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.
%H A386255 Alois P. Heinz, <a href="/A386255/b386255.txt">Table of n, a(n) for n = 0..650</a>
%F A386255 E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / (j-1)!).
%e A386255 a(3) = 15 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2), (1#,2,2#), (2#,1#,2), (2,1#,2#), (2#,2,1#), (2,2#,1#), (2#,2,2), (2,2#,2), (2,2,2#), (3#,3,3), (3,3#,3), (3,3,3#) where # denotes a mark.
%p A386255 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A386255       b(n, i-1)+add(b(n-j, min(n-j, i-1))/(j-1)!, j=i..n)))
%p A386255     end:
%p A386255 a:= n-> n!*b(n$2):
%p A386255 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jul 17 2025
%t A386255 terms=24; CoefficientList[Series[Product[1+Sum[x^j/(j-1)!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* _Stefano Spezia_, Jul 17 2025 *)
%o A386255 (PARI) E_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/((i-1)!)))))}
%Y A386255 Cf. A002999, A005183, A007837, A032299, A347006, A386254.
%K A386255 nonn,easy
%O A386255 0,3
%A A386255 _John Tyler Rascoe_, Jul 16 2025