A331672 Sum of all base-n numbers with digit sum n and length at most n.

3, 91, 2635, 94501, 4254936, 234572213, 15403880115, 1176838159861, 102631111100848, 10063085278250005, 1095923297151849530, 131253123286275198027, 17145216226230367266330, 2425892898650501790637545, 369599184391990522425455939, 60326656013944234430010524773

Offset: 2

Author

Alois P. Heinz, Feb 22 2020

Keywords

• nonn
• base

Comments

The cardinality of these numbers is given by A048775(n-1).

Examples

			a(2) = 3 = 11_2.
a(3) = 91 = 5 + 7 + 11 + 13 + 15 + 19 + 21 = 12_3 + 21_3 + 102_3 + 111_3 + 120_3 + 201_3 + 210_3.
a(10) = A130835(10) = 102631111100848.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..322

Crossrefs

Cf. A007953, A023037, A048775, A130835, A331673.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
      `if`(i=0, 0, add((p->[p[1], p[2]*k+p[1]*d])(
         b(n-d, i-1, k)), d=0..min(n, k-1))))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=2..17);
# second Maple program:
a:= n-> (binomial(2*n-1, n)-n)*(n^n-1)/(n-1):
seq(a(n), n=2..17);

Formula

a(n) = A048775(n-1)*A023037(n) = (binomial(2*n-1,n)-n)*(n^n-1)/(n-1).