A167403 Number of decimal numbers having n or fewer digits and having the sum of their digits equal to n.

1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92368, 352595, 1351142, 5194385, 20024980, 77384340, 299671971, 1162635441, 4518099300, 17583582225, 68522664400, 267350823015, 1044243559263, 4082760176300, 15977236602150, 62576817828876, 245279492151021

Offset

1,2

Comments

a(3) = 10, because 10 decimal numbers have 3 or fewer digits and a digit sum of 3: 3, 30, 300, 12, 120, 201, 21, 210, 102, 111.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1665

J. M. Shunia and L. Sauras Altuzarra, Arithmetic terms for sums of multinomial coefficients, Ramanujan Journal, vol. 68, 2025.

Formula

a(n) = [x^n] ((x^10-1)/(x-1))^n.

a(n) = floor(((10^(10*n)-1)/(100^n-10^n))^n) mod 10^n, n > 0 (see Theorem 5.1 from our article). - Joseph M. Shunia and Lorenzo Sauras Altuzarra, Mar 27 2026

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i=0, 0,
       add(b(n-j, i-1), j=0..min(n, 9)) ))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..30);

Crossrefs

Cf. A130835, A001700.

Column k=9 of A305161.

Sequence in context: A363781 A318115 A318116 * A318117 A001700 A088218

Adjacent sequences: A167400 A167401 A167402 * A167404 A167405 A167406

Keyword

base,easy,nonn

Author

Alois P. Heinz, Nov 02 2009

Status

approved