A206817 Sum_{0<j<k<=n} (k!-j!).

1, 10, 73, 520, 3967, 33334, 309661, 3166468, 35416555, 430546642, 5655609529, 79856902816, 1206424711303, 19419937594990, 331860183278677, 6000534640290364, 114462875817046051, 2297294297649673738, 48394006967070653425

Offset

2,2

Comments

In the following guide to related sequences,

c(n) = Sum_{0<j<n} s(n)-s(j),

t(n) = Sum_{0<j<k<=n} s(k)-s(j).

s(k).................c(n)........t(n)

k....................A000217.....A000292

k^2..................A016061.....A004320

k^3..................A206808.....A206809

k^4..................A206810.....A206811

k!...................A206816.....A206817

prime(k).............A152535.....A062020

prime(k+1)...........A185382.....A206803

2^(k-1)..............A000337.....A045618

k(k+1)/2.............A007290.....A034827

k-th quarter-square..A049774.....A206806

Links

Danny Rorabaugh, Table of n, a(n) for n = 2..400

Formula

a(n) = a(n-1)+(n-1)s(n)-p(n-1), where s(n) = n! and p(k) = 1!+2!+...+k!.

a(n) = Sum_{k=2..n} A206816(k).

Example

a(3) = (2-1) + (6-1) + (6-2) = 10.

Mathematica

s[k_] := k!; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1];
Table[c[n], {n, 2, 32}]          (* A206816 *)
Flatten[Table[t[n], {n, 2, 20}]] (* A206817 *)

Prog

(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
(PARI) a(n)=sum(j=1, n, j!*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015
(PARI) a(n)=my(t=1); sum(j=1, n, t*=j; t*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015

Crossrefs

Cf. A000142, A206816.

Sequence in context: A243878 A200580 A181678 * A159687 A199556 A044197

Adjacent sequences: A206814 A206815 A206816 * A206818 A206819 A206820

Keyword

nonn

Author

Clark Kimberling, Feb 12 2012

Status

approved