A386254 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.

1, 1, 2, 6, 18, 60, 240, 1085, 5012, 23730, 121440, 685707, 4144668, 25614589, 159141892, 1012740885, 6805631232, 48872707006, 369227821608, 2853779791619, 22131042288980, 172055270717463, 1362017827326860, 11208504802237327, 96939147303239304, 875473007351905045

Offset: 0

Author

John Tyler Rascoe, Jul 16 2025

Keywords

• nonn
• easy

Examples

			a(3) = 6 counts: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2), (3,3,3).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..674

Crossrefs

Cf. A002999, A005183, A007837, A032299, A347006, A386255.

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!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 17 2025

Mathematica

terms=26; CoefficientList[Series[Product[1+Sum[x^j/j!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

PARI

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

Formula

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