A386255 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 and exactly one of each kind of letter is marked.

1, 1, 4, 15, 64, 325, 1776, 11179, 72640, 489969, 3435580, 26495491, 221599104, 1893705697, 16145571820, 138299146665, 1241234863936, 12033569772769, 124055067568788, 1303750295285563, 13577876900409280, 139418829477000801, 1441311794301705964, 15537427948684769425

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..650

Formula

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / (j-1)!).

Example

a(3) = 15 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2), (1#,2,2#), (2#,1#,2), (2,1#,2#), (2#,2,1#), (2,2#,1#), (2#,2,2), (2,2#,2), (2,2,2#), (3#,3,3), (3,3#,3), (3,3,3#) where # denotes a mark.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/(j-1)!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 17 2025

Mathematica

terms=24; CoefficientList[Series[Product[1+Sum[x^j/(j-1)!, {j, k, terms}], {k, terms}], {x, 0, terms-1}], x]Range[0, terms-1]! (* Stefano Spezia, Jul 17 2025 *)

Prog

(PARI) E_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1, N, 1 + sum(i=k, N, x^i/((i-1)!)))))}

Crossrefs

Cf. A002999, A005183, A007837, A032299, A347006, A386254.

Sequence in context: A117669 A323789 A341922 * A007526 A233536 A349202

Adjacent sequences: A386252 A386253 A386254 * A386256 A386257 A386258

Keyword

nonn,easy

Author

John Tyler Rascoe, Jul 16 2025

Status

approved