A397684 Number of n-digit numbers such that, if the digit j occurs, then every digit from 0 to j also occurs.

0, 1, 7, 49, 391, 3601, 37927, 451249, 5995591, 88073041, 1381849447, 22012209649, 342462451591, 5107803726481, 72689981888167, 989028944467249, 12924509604552391, 163044766928275921, 1995220493407066087, 23786745463088040049, 277301139700968681991, 3171128127031352793361

Offset

1,3

Comments

Equivalently, a(n) is the number of n-digit numbers whose set of decimal digits is {0,1,...,k} for some k, 1 <= k <= 9.

a(n) is the number of n-digit terms of A305701.

For 1 <= n <= 10, a(n) = A343583(n-1). The sequences first differ at n = 11, since decimal notation has only the ten digits 0,1,...,9.

Links

Table of n, a(n) for n=1..22.

Formula

a(n) = Sum_{k=1..9} k*k!*StirlingS2(n,k+1).

Example

a(3) = 7, since the 3-digit numbers satisfying the condition are 100, 101, 102, 110, 120, 201, and 210.

Mathematica

a[n_] := Sum[k*k!*StirlingS2[n, k + 1], {k, 1, Min[9, n - 1]}];
Table[a[n], {n, 1, 22}]

Crossrefs

Cf. A305701 (the numbers being counted), A343583.

Sequence in context: A349781 A199554 A383958 * A343583 A221462 A374855

Adjacent sequences: A397680 A397681 A397682 * A397686 A397687 A397688

Keyword

nonn,base

Author

Bartlomiej Pawlik, Jul 05 2026

Status

approved