A055555 -id:A055555 - cp's OEIS frontend

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

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

Offset: 1

R. J. Cano, Mar 01 2013

			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.

R. J. Cano, Table of n, a(n) for n = 1..60
R. J. Cano, Additional information

Cf. A181482, A055555.

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

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

Olivier Gérard, Oct 24 2017

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

			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.

Iain Fox, Table of n, a(n) for n = 0..252

Cf. A000217 (triangular numbers).
Cf. A001563 (difference of factorials).
Cf. A049775 (same idea between consecutive powers of 2).
Cf. A000142, A055555.

Table[1/2 ((n + 1)! ((n + 1)! + 1) - n! (n! + 1) ), {n, 0, 10}]

a(n) = 1/2*((n+1)!*((n+1)! + 1)-n!*(n!+1)) \\ Iain Fox, Nov 28 2017

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

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

Simon Plouffe

Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A000142 (n!), A055555.

[Factorial(n)*(Factorial(n)-1): n in [0..30]]; // Vincenzo Librandi, Apr 15 2015

Table[n! (n! - 1), {n, 0, 20}] (* Vincenzo Librandi, Apr 15 2015 *)

vector(20, n, n--; (n!)^2-n!) \\ Michel Marcus, Apr 15 2015

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

Jason Earls, Aug 07 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..60

Cf. A055555.

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

Kelvin Voskuijl, Aug 12 2025

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

			2.3844273879714288211644804923804481846149857064...

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

evalf(sum(2/(n!*(n!+1)),n=0..infinity), 120);  # Alois P. Heinz, Aug 13 2025

PARI
```
suminf(k=0, 2/(k!*(k!+1)))
```

sumpos(k=0,1/binomial(k!+1,2)) \\ Charles R Greathouse IV, Aug 19 2025

Equals Sum_{k>=0} 1/A055555(k).

cp's OEIS Frontend

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

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

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

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Magma

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

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

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