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.
1, 1, 1, 4, 17, 96, 652, 5356, 51361, 568840, 7157036, 101048454, 1582644956, 27224336244, 509883010652, 10319902635984, 224283040843745, 5205554049801528, 128430045368430484, 3354764715348964222, 92460461868234201532, 2680680433302859375630, 81542551486359310209666
Offset: 0
Examples
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].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..425
Programs
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Maple
b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)/j!, j=1..min(n, t))) end: a:= n-> n!*add(b(n-j, j-1)/j!, j=0..n): seq(a(n), n=0..22); # Alois P. Heinz, Jul 19 2025 -
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!))))))}
Formula
E.g.f.: Sum_{i>=0} x^i/(i! * (1 - Sum_{j=1..i-1} x^j/j!)).