cp's OEIS Frontend

A386375 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs more frequently than any other letter.

%I A386375 #12 Jul 27 2025 00:20:27
%S A386375 1,1,1,4,17,96,652,5356,51361,568840,7157036,101048454,1582644956,
%T A386375 27224336244,509883010652,10319902635984,224283040843745,
%U A386375 5205554049801528,128430045368430484,3354764715348964222,92460461868234201532,2680680433302859375630,81542551486359310209666
%N A386375 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs more frequently than any other letter.
%H A386375 Alois P. Heinz, <a href="/A386375/b386375.txt">Table of n, a(n) for n = 0..425</a>
%F A386375 E.g.f.: Sum_{i>=0} x^i/(i! * (1 - Sum_{j=1..i-1} x^j/j!)).
%e A386375 a(5) = 96 counts the following words (number of permutations shown in brackets): (1,1,1,1,1) [1], (1,1,1,1,2) [5], (1,1,1,2,2) [10], (1,1,1,2,3) [20], (1,1,2,3,4) [60].
%p A386375 b:= proc(n, t) option remember; `if`(n=0, 1,
%p A386375       add(b(n-j, t)/j!, j=1..min(n, t)))
%p A386375     end:
%p A386375 a:= n-> n!*add(b(n-j, j-1)/j!, j=0..n):
%p A386375 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jul 19 2025
%o A386375 (PARI) B_x(N) = {my(x='x+O('x^N)); Vec(serlaplace( sum(i=0,N, x^i/(i!*(1-sum(j=1,i-1, x^j/j!))))))}
%Y A386375 Cf. A000262, A000670, A006153, A308876, A386374.
%K A386375 nonn,easy
%O A386375 0,4
%A A386375 _John Tyler Rascoe_, Jul 19 2025