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A206817 Sum_{0

%I A206817 #21 Oct 11 2015 02:55:02
%S A206817 1,10,73,520,3967,33334,309661,3166468,35416555,430546642,5655609529,
%T A206817 79856902816,1206424711303,19419937594990,331860183278677,
%U A206817 6000534640290364,114462875817046051,2297294297649673738,48394006967070653425
%N A206817 Sum_{0<j<k<=n} (k!-j!).
%C A206817 In the following guide to related sequences,
%C A206817 c(n) = Sum_{0<j<n} s(n)-s(j),
%C A206817 t(n) = Sum_{0<j<k<=n} s(k)-s(j).
%C A206817 s(k).................c(n)........t(n)
%C A206817 k....................A000217.....A000292
%C A206817 k^2..................A016061.....A004320
%C A206817 k^3..................A206808.....A206809
%C A206817 k^4..................A206810.....A206811
%C A206817 k!...................A206816.....A206817
%C A206817 prime(k).............A152535.....A062020
%C A206817 prime(k+1)...........A185382.....A206803
%C A206817 2^(k-1)..............A000337.....A045618
%C A206817 k(k+1)/2.............A007290.....A034827
%C A206817 k-th quarter-square..A049774.....A206806
%H A206817 Danny Rorabaugh, <a href="/A206817/b206817.txt">Table of n, a(n) for n = 2..400</a>
%F A206817 a(n) = a(n-1)+(n-1)s(n)-p(n-1), where s(n) = n! and p(k) = 1!+2!+...+k!.
%F A206817 a(n) = Sum_{k=2..n} A206816(k).
%e A206817 a(3) = (2-1) + (6-1) + (6-2) = 10.
%t A206817 s[k_] := k!; t[1] = 0;
%t A206817 p[n_] := Sum[s[k], {k, 1, n}];
%t A206817 c[n_] := n*s[n] - p[n];
%t A206817 t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1];
%t A206817 Table[c[n], {n, 2, 32}]          (* A206816 *)
%t A206817 Flatten[Table[t[n], {n, 2, 20}]] (* A206817 *)
%o A206817 (Sage) [sum([sum([factorial(k)-factorial(j) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,21)] # _Danny Rorabaugh_, Apr 18 2015
%o A206817 (PARI) a(n)=sum(j=1,n,j!*(2*j-n-1)) \\ _Charles R Greathouse IV_, Oct 11 2015
%o A206817 (PARI) a(n)=my(t=1); sum(j=1,n,t*=j; t*(2*j-n-1)) \\ _Charles R Greathouse IV_, Oct 11 2015
%Y A206817 Cf. A000142, A206816.
%K A206817 nonn
%O A206817 2,2
%A A206817 _Clark Kimberling_, Feb 12 2012