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A206816 a(n) = Sum_{0

%I A206816 #20 Jun 15 2025 10:00:19
%S A206816 1,9,63,447,3447,29367,276327,2856807,32250087,395130087,5225062887,
%T A206816 74201293287,1126567808487,18213512883687,312440245683687,
%U A206816 5668674457011687,108462341176755687,2182831421832627687,46096712669420979687
%N A206816 a(n) = Sum_{0<j<n} (n!-j!).
%H A206816 Danny Rorabaugh, <a href="/A206816/b206816.txt">Table of n, a(n) for n = 2..400</a>
%F A206816 a(n) = n*n!-p(n), where p(n) is the n-th partial sum of (j!).
%F A206816 a(n) = t(n)-t(n-1), where t = A206817.
%F A206816 a(n) = Sum_{k=1..n-1} k^2*k!. - _Ridouane Oudra_, Jun 13 2025
%F A206816 a(n) = A001563(n) - A007489(n). - _Ridouane Oudra_, Jun 14 2025
%e A206816 a(4) = (24-1) + (24-2) + (24-6) = 63.
%p A206816 seq(add(k^2*k!,k=1..n-1), n=2..30); # _Ridouane Oudra_, Jun 13 2025
%t A206816 s[k_] := k!; t[1] = 0;
%t A206816 p[n_] := Sum[s[k], {k, 1, n}];
%t A206816 c[n_] := n*s[n] - p[n];
%t A206816 t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1];
%t A206816 Table[c[n], {n, 2, 32}]           (* A206816 *)
%t A206816 Flatten[Table[t[n], {n, 2, 20}]]  (* A206817 *)
%o A206816 (Sage) [sum([factorial(n)-factorial(j) for j in range(1,n)]) for n in range(2,21)] # _Danny Rorabaugh_, Apr 18 2015
%o A206816 (PARI) a(n) = sum(j=1, n-1, n!-j!); \\ _Michel Marcus_, Jun 13 2025
%Y A206816 Cf. A000142, A206817, A001563, A007489.
%K A206816 nonn
%O A206816 2,2
%A A206816 _Clark Kimberling_, Feb 12 2012