A347006 E.g.f.: Product_{k>=1} (1 + exp(x) * x^k / k!).

1, 1, 3, 10, 43, 206, 1044, 5909, 38371, 272314, 1995208, 14869889, 115433344, 965259881, 8773348601, 84608514095, 837220780691, 8334354200226, 83498917650084, 855936118936073, 9180736840445788, 104439240481045949, 1253608634906635901

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

• nonn
• easy

Comments

a(n) is the number of words of length n over an infinite alphabet such that for any letter k appearing within a word, exactly k occurrences of that letter are marked. - John Tyler Rascoe, Jul 16 2025

Examples

			a(3) = 10 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2#), (2#,1#,2#), (2#,2#,1#), (2#,2#,2), (2#,2,2#), (2,2#,2#), (3#,3#,3#). - _John Tyler Rascoe_, Jul 16 2025
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..612 (first 101 terms from John Tyler Rascoe)

Crossrefs

Cf. A007837, A265952, A305547, A347005.

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))/i!/(j-i)!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 17 2025

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

PARI

C_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + exp(x)*x^k/k!)))} \\ John Tyler Rascoe, Jul 16 2025

Formula

E.g.f.: exp( Sum_{k>=1} ( Sum_{d|k} (-1)^(d+1) * exp(d*x) / (d * ((k/d)!)^d) ) * x^k ).

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