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

%I A332711 #30 Sep 29 2023 14:09:34
%S A332711 0,1,5,28,203,1936,23517,349504,6149495,124999936,2881935953,
%T A332711 74300836864,2118007738035,66142897770496,2245609694259557,
%U A332711 82351536043343872,3244079458377786863,136619472483668525056,6125138252818308310041,291271111111111111081984
%N A332711 Sum of all numbers in bijective base-n numeration with digit sum n.
%C A332711 The number of numbers in bijective base-n numeration with digit sum n equals the number of compositions of n: A000079(n).
%H A332711 Alois P. Heinz, <a href="/A332711/b332711.txt">Table of n, a(n) for n = 0..387</a>
%H A332711 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bijective_numeration">Bijective numeration</a>
%H A332711 Wikipedia, <a href="https://en.wikipedia.org/wiki/Digit_sum">Digit sum</a>
%F A332711 a(n) = ((n+1)^n - 2^n) / (n - 1) for n >= 2. - _Peter Bala_, Sep 28 2023
%e A332711 a(0) =  0.
%e A332711 a(1) =  1 = 1_bij1.
%e A332711 a(2) =  5 = 3 + 2 = 11_bij2 + 2_bij2.
%e A332711 a(3) = 28 = 13 + 7 + 5 + 3 = 111_bij3 + 21_bij3 + 12_bij3 + 3_bij3.
%p A332711 b:= proc(n, k) option remember; `if`(n=0, [1, 0], add((p->
%p A332711       [p[1], p[2]*k+p[1]*d])(b(n-d, k)), d=1..min(n, k)))
%p A332711     end:
%p A332711 a:= n-> b(n$2)[2]:
%p A332711 seq(a(n), n=0..23);
%t A332711 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]}]];
%t A332711 a[n_] := b[n, n][[2]];
%t A332711 a /@ Range[0, 23] (* _Jean-François Alcover_, Apr 23 2021, after _Alois P. Heinz_ *)
%Y A332711 Cf. A000079, A007953, A214676.
%K A332711 nonn
%O A332711 0,3
%A A332711 _Alois P. Heinz_, Feb 20 2020