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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..425

Formula

E.g.f.: Sum_{i>=0} x^i/(i! * (1 - Sum_{j=1..i-1} x^j/j!)).

Example

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].

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

Prog

(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!))))))}

Crossrefs

Cf. A000262, A000670, A006153, A308876, A386374.

Sequence in context: A278644 A249078 A385125 * A353546 A024052 A128321

Adjacent sequences: A386372 A386373 A386374 * A386376 A386377 A386378

Keyword

nonn,easy

Author

John Tyler Rascoe, Jul 19 2025

Status

approved