cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A206817 Sum_{0

Original entry on oeis.org

1, 10, 73, 520, 3967, 33334, 309661, 3166468, 35416555, 430546642, 5655609529, 79856902816, 1206424711303, 19419937594990, 331860183278677, 6000534640290364, 114462875817046051, 2297294297649673738, 48394006967070653425
Offset: 2

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

In the following guide to related sequences,
c(n) = Sum_{0
t(n) = Sum_{0
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

Examples

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

Links

Crossrefs

Cf. A000142, A206816.

Programs

  • 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 *)
  • 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
  • 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

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).

A185009 Row sums of A051949 (differences of factorial numbers), seen as a triangle.

Original entry on oeis.org

0, 5, 45, 351, 2847, 25047, 241047, 2534247, 28984167, 358842087, 4785978087, 68453274087, 1045616538087, 16993016806887, 292825130163687, 5333909818803687, 102415654899123687, 2067588695129523687, 43785455761653171687, 970599475776544179687
Offset: 1

Views

Author

Olivier Gérard, Nov 02 2012

Keywords

Links

Crossrefs

cf. A051949.
Other summations of differences of factorials : A206816, A206817, A065355.

Programs

  • Mathematica
    Table[Plus @@ Prepend[Table[(n + 1)! - i!, {i, n, 2, -1}], (n)! - 1], {n, 0, 20}]
  • PARI
    for(n=1,25, print1((n^2-1)*n! - sum(k=1,n-1, k!), ", ")) \\ G. C. Greubel, Jun 09 2017

Formula

a(n)= (n-1)*(n+1)*n! - sum( i!, i=1..n-1)

A051949 Differences of two factorial numbers.

Original entry on oeis.org

0, 1, 4, 5, 18, 22, 23, 96, 114, 118, 119, 600, 696, 714, 718, 719, 4320, 4920, 5016, 5034, 5038, 5039, 35280, 39600, 40200, 40296, 40314, 40318, 40319, 322560, 357840, 362160, 362760, 362856, 362874, 362878, 362879, 3265920
Offset: 0

Views

Author

N. J. A. Sloane, Jan 05 2000

Keywords

Comments

Contribution from Olivier Gérard, Oct 09 2012: (Start)
This sequence can also be seen as a triangular table starting at offset 1.
0
1
4 5
18 22 23
96 114 118 119
600 696 714 718 719
4320 4920 5016 5034 5038 5039
First column is A001563, first diagonal is A033312, row sum is A206816.
;
or starting at offset 0:
0
1 4
5 18 22
23 96 114 118
119 600 696 714 718
719 4320 4920 5016 5034 5038
First column is A033312, first diagonal is n!-2, row sum is A185009.
(End)

Crossrefs

Cf. A204930 (ordered differences of factorial).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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