A392965 Least number of vertices in a stepwise-irregular tree with at least one vertex of degree n.

1, 2, 3, 7, 21, 81, 391, 2283, 15657, 123301, 1096011, 10850511, 118369213, 1410566457, 18228858831, 253901962291, 3791602636881, 60428667025293, 1023732711957907, 18370314775689111, 348069122065688421, 6943978985210484001, 145492893023457760023

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Ivan Gutman, Stepwise irregular graphs, Applied Mathematics and Computation 325 (2018), 234-238.

Wikipedia, Stepwise irregular graph.

Formula

a(n) = 1 + n * Sum_{k=1..n} Product_{i=2..k} (n-i).

a(n) = 1 + n * Sum_{k=0..n-2} (n-2)!/k! for n>=2.

a(n) = 1 + n * A000522(n-2) for n>=2. - Alois P. Heinz, Jan 28 2026

Maple

a:= proc(n) option remember; `if`(n<2, n+1,
      ((n-2)*(2*n+1)*a(n-1)-(2*n-1)*(n-3)*a(n-2))/(2*n-3))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jan 28 2026

Prog

(Python)
from math import factorial
def a(n):
  if n >= 0 and n <= 2: return n + 1
  scale_num = factorial(n - 2)
  sum_val = 0
  for i in range(n - 1):
    sum_val += scale_num // factorial(i)
  return 1 + n * sum_val

Crossrefs

Cf. A000522.

Sequence in context: A107108 A323230 A393926 * A262305 A189360 A001532

Adjacent sequences: A392962 A392963 A392964 * A392966 A392967 A392968

Keyword

nonn,easy

Author

Kyle Wood, Jan 28 2026

Status

approved