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A332082 a(n) = Sum_{1 <= m <= n} Sum_{1 <= k <= n+1-m} m*R(k,n+1), where R(k,b) = (b^k - 1)/(b - 1) is the base-b repunit of length k.

%I A332082 #16 Sep 19 2020 23:57:44
%S A332082 0,1,7,42,295,2675,31122,447188,7661370,152415765,3452271185,
%T A332082 87693358654,2468455488681,76256200336407,2564715882332660,
%U A332082 93281313241869480,3647955866777821668,152635100350763019705,6803550294289868214315,321844061970058547739730,16103630469426364324556635
%N A332082 a(n) = Sum_{1 <= m <= n} Sum_{1 <= k <= n+1-m} m*R(k,n+1), where R(k,b) = (b^k - 1)/(b - 1) is the base-b repunit of length k.
%C A332082 This is the sum of all elements of a triangular matrix (of size n given by the index) where the k-th diagonal is filled with k-repdigits (in base n+1, as to have digits up to n) of increasing length, as in
%C A332082   ( 1   0    0    . . . . . . . .  0   )
%C A332082   ( 2  11    0                     :   )
%C A332082   ( 3  22   111   ˙ · .            :   )
%C A332082   ( :  33   222   ˙ · .  ˙ · .     :   )
%C A332082   ( :      ˙ · .  ˙ · .  ˙ · .     0   )
%C A332082   ( n  . . . . .   3..3  2...2  1....1 )
%C A332082 with numbers written in base n+1.
%e A332082 a(2) = 2 + 1 + 11[3] = 3 + 4 = 7.
%e A332082 a(3) = 3 + 2 + 22[4] + 1 + 11[4] + 111[4] = 6 + 15 + 21 = 42.
%e A332082 a(4) = 4 + 3 + 33[5] + 2 + 22[5] + 222[5] + 1 + 11[5] + 111[5] + 1111[5] = 10 + 36 + 93 + 156 = 295.
%t A332082 a[n_] := Sum[(n + 1 - m)*((n + 1)*((n + 1)^m - 1) - m*n)/n^2, {m, 1, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Aug 24 2020 *)
%o A332082 (PARI) apply( {A332082(n)=sum(m=1,n,(n+1-m)*((n+1)*((n+1)^m-1)-m*n)\n^2)}, [0..20])
%Y A332082 For base 10 repunits and repdigits, cf. A002275 (repunits), A010785 (repdigits) and A014824 (partial sums of repunits).
%K A332082 nonn,base
%O A332082 0,3
%A A332082 _M. F. Hasler_, Aug 19 2020