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

Author

John Tyler Rascoe, Jul 16 2025

Keywords

• nonn
• easy

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)

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)!)))))}

Formula

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