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A386254 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.

%I A386254 #16 Jul 18 2025 04:30:46
%S A386254 1,1,2,6,18,60,240,1085,5012,23730,121440,685707,4144668,25614589,
%T A386254 159141892,1012740885,6805631232,48872707006,369227821608,
%U A386254 2853779791619,22131042288980,172055270717463,1362017827326860,11208504802237327,96939147303239304,875473007351905045
%N A386254 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.
%H A386254 Alois P. Heinz, <a href="/A386254/b386254.txt">Table of n, a(n) for n = 0..674</a>
%F A386254 E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / j!).
%e A386254 a(3) = 6 counts: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2), (3,3,3).
%p A386254 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A386254       b(n, i-1)+add(b(n-j, min(n-j, i-1))/j!, j=i..n)))
%p A386254     end:
%p A386254 a:= n-> n!*b(n$2):
%p A386254 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 17 2025
%t A386254 terms=26; CoefficientList[Series[Product[1+Sum[x^j/j!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* _Stefano Spezia_, Jul 17 2025 *)
%o A386254 (PARI) D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/i!))))}
%Y A386254 Cf. A002999, A005183, A007837, A032299, A347006, A386255.
%K A386254 nonn,easy
%O A386254 0,3
%A A386254 _John Tyler Rascoe_, Jul 16 2025