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 1-5 of 5 results.

A213203 The sum of the first n! integers, with every n-th integer taken as negative.

Original entry on oeis.org

-1, -1, 3, 132, 4260, 172440, 9069480, 609618240, 51209444160, 5267273961600, 651825357321600, 95601055094899200, 16405141092269529600, 3257166195621552614400, 741005309513165913216000
Offset: 1

Views

Author

R. J. Cano, Mar 01 2013

Keywords

Examples

			For a(3)=3, 3! is 6 then the sum of the first 6 integers taking each 3rd integer as negative is: 1+2-3+4+5-6 = 3.
For a(4)=132, 4! is 24 then the sum of the first 24 integers taking each 4th integer as negative is: 1+2+3-4+5+6+7-8+9+10+11-12+13+14+15-16+17+18+19-20+21+22+23-24 = 132.
		

Crossrefs

Programs

  • PARI
    a(n)={my(y=(n-1)!);((n*y)*((n-2)*y-1))\2;}

Formula

a(n) = n * (n-1)! * ((n-2)*(n-1)! - 1)/2.
Conjecture: a(n) + (-n^2-n-11)*a(n-1) + (n^3+7*n^2-13*n+39)*a(n-2) - 2*(n-2)*(4*n^2-2*n-15)*a(n-3) + 20*(n-2)*(n-3)*(n-4)*a(n-4) = 0. - R. J. Mathar, Mar 21 2013

A294193 a(n) = sum of integers between n!+1 and (n+1)!.

Original entry on oeis.org

0, 2, 18, 279, 6960, 252300, 12443760, 800168040, 65028257280, 6518255405760, 790091384544000, 113924591159702400, 19273172758289049600, 3780639334294658035200, 851206099134433961318400, 218026562222345234117760000, 63037891684425054948655104000
Offset: 0

Views

Author

Olivier Gérard, Oct 24 2017

Keywords

Comments

Useful as a growth reference for sequences summing on intervals between 2 factorials.

Examples

			a(2) = 3 + 4 + 5 + 6 = 18.
a(3) = 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 = 24*25/2 - 6*7/2 = 279.
		

Crossrefs

Cf. A000217 (triangular numbers).
Cf. A001563 (difference of factorials).
Cf. A049775 (same idea between consecutive powers of 2).

Programs

  • Mathematica
    Table[1/2 ((n + 1)! ((n + 1)! + 1) - n! (n! + 1) ), {n, 0, 10}]
  • PARI
    a(n) = 1/2*((n+1)!*((n+1)! + 1)-n!*(n!+1)) \\ Iain Fox, Nov 28 2017

Formula

a(n) = (1/2) * ((n + 1)!*((n + 1)! + 1) - n!*(n! + 1) ).
a(n) = A055555(n+1) - A055555(n). - J.S. Seneschal, Jul 07 2025

Extensions

More terms from Iain Fox, Nov 28 2017

A020547 2nd Euler polynomial x^2 - x evaluated at x=n!.

Original entry on oeis.org

0, 0, 2, 30, 552, 14280, 517680, 25396560, 1625662080, 131681531520, 13168185811200, 1593350882323200, 229442532323558400, 38775788037405619200, 7600054456464819148800, 1710012252722891749632000, 437763136697374129754112000
Offset: 0

Views

Author

Keywords

Crossrefs

Cf. A000142 (n!), A055555.

Programs

Formula

a(n) = n!*(n!-1). - Mark van Hoeij, May 13 2013
a(n) = A000142(n)*A000142(n-1). - Michel Marcus, Apr 15 2015
a(n) = 2*(A055555(n) - A000142(n)). - J.S. Seneschal, Jul 01 2025

A063101 a(n) is the number of divisors of n!*(n! + 1)/2.

Original entry on oeis.org

1, 1, 2, 4, 18, 36, 96, 144, 336, 1120, 960, 960, 4320, 5760, 9504, 29568, 161280, 80640, 884736, 110592, 155520, 460800, 729600, 2188800, 929280, 1300992, 2040192, 1298304, 3528000, 14112000, 71884800, 71884800, 85708800, 243793920
Offset: 0

Views

Author

Jason Earls, Aug 07 2001

Keywords

Crossrefs

Cf. A055555.

Programs

  • PARI
    a(n)={numdiv(n!*(n! + 1)/2)}

A386990 Decimal expansion of Sum_{k>=0} 2/(k!*(k! + 1)).

Original entry on oeis.org

2, 3, 8, 4, 4, 2, 7, 3, 8, 7, 9, 7, 1, 4, 2, 8, 8, 2, 1, 1, 6, 4, 4, 8, 0, 4, 9, 2, 3, 8, 0, 4, 4, 8, 1, 8, 4, 6, 1, 4, 9, 8, 5, 7, 0, 6, 4, 6, 6, 9, 8, 7, 8, 4, 8, 4, 1, 7, 2, 0, 3, 9, 5, 2, 0, 8, 9, 0, 0, 3, 8, 3, 7, 7, 6, 3, 0, 4, 4, 7, 1, 1, 5, 3, 9, 1, 3, 2, 1, 6, 2, 4, 2, 6, 7, 8, 5, 5, 9, 3, 9, 6, 9, 5, 2, 3
Offset: 1

Views

Author

Kelvin Voskuijl, Aug 12 2025

Keywords

Comments

Sum of reciprocals of A055555 (triangular numbers of factorials).

Examples

			2.3844273879714288211644804923804481846149857064...
		

Crossrefs

Cf. A000217, A070910 (of n!^2), A055555, A091131 (of n!).

Programs

  • Maple
    evalf(sum(2/(n!*(n!+1)),n=0..infinity), 120);  # Alois P. Heinz, Aug 13 2025
  • PARI
    suminf(k=0, 2/(k!*(k!+1)))
    
  • PARI
    sumpos(k=0,1/binomial(k!+1,2)) \\ Charles R Greathouse IV, Aug 19 2025

Formula

Equals Sum_{k>=0} 1/A055555(k).
Showing 1-5 of 5 results.