A332711 Sum of all numbers in bijective base-n numeration with digit sum n.

0, 1, 5, 28, 203, 1936, 23517, 349504, 6149495, 124999936, 2881935953, 74300836864, 2118007738035, 66142897770496, 2245609694259557, 82351536043343872, 3244079458377786863, 136619472483668525056, 6125138252818308310041, 291271111111111111081984

Offset

0,3

Comments

The number of numbers in bijective base-n numeration with digit sum n equals the number of compositions of n: A000079(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..387

Wikipedia, Bijective numeration

Wikipedia, Digit sum

Formula

a(n) = ((n+1)^n - 2^n) / (n - 1) for n >= 2. - Peter Bala, Sep 28 2023

Example

a(0) =  0.
a(1) =  1 = 1_bij1.
a(2) =  5 = 3 + 2 = 11_bij2 + 2_bij2.
a(3) = 28 = 13 + 7 + 5 + 3 = 111_bij3 + 21_bij3 + 12_bij3 + 3_bij3.

Maple

b:= proc(n, k) option remember; `if`(n=0, [1, 0], add((p->
      [p[1], p[2]*k+p[1]*d])(b(n-d, k)), d=1..min(n, k)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..23);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*k + p[[1]]*d}][b[n - d, k]], {d, 1, Min[n, k]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 23] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000079, A007953, A214676.

Sequence in context: A216586 A151500 A356409 * A292426 A347005 A356025

Adjacent sequences: A332708 A332709 A332710 * A332712 A332713 A332714

Keyword

nonn

Author

Alois P. Heinz, Feb 20 2020

Status

approved