cp's OEIS Frontend

A386374 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs at least as many times as any other letter.

%I A386374 #16 Jul 27 2025 00:20:21
%S A386374 1,1,3,10,47,276,2022,17606,179391,2093860,27581888,404680398,
%T A386374 6541528886,115437202986,2206844818622,45408726154590,
%U A386374 1000134868827263,23468606700087972,584340284516996400,15383829737201853518,426915367401366308112,12454073547413511363878
%N A386374 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs at least as many times as any other letter.
%H A386374 Alois P. Heinz, <a href="/A386374/b386374.txt">Table of n, a(n) for n = 0..425</a>
%F A386374 E.g.f.: Sum_{i>=0} x^i/(i! * (1 - Sum_{j=1..i} x^j/j!)).
%e A386374 a(3) = 10 counts: (1,1,1), (1,1,2), (1,2,1), (1,2,3), (1,3,2), (2,1,1), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
%p A386374 b:= proc(n, t) option remember; `if`(n=0, 1,
%p A386374       add(b(n-j, t)/j!, j=1..min(n, t)))
%p A386374     end:
%p A386374 a:= n-> n!*add(b(n-j, j)/j!, j=0..n):
%p A386374 seq(a(n), n=0..21);  # _Alois P. Heinz_, Jul 19 2025
%o A386374 (PARI) A_x(N) = {my(x='x+O('x^N)); Vec(serlaplace(sum(i=0,N, x^i/(i! *(1-sum(j=1,i, x^j/j!))))))}
%Y A386374 Cf. A000262, A000670, A006153, A308876.
%K A386374 nonn,easy
%O A386374 0,3
%A A386374 _John Tyler Rascoe_, Jul 19 2025