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.

A007911 a(n) = (n-1)!! - (n-2)!!.

Original entry on oeis.org

1, 1, 5, 7, 33, 57, 279, 561, 2895, 6555, 35685, 89055, 509985, 1381905, 8294895, 24137505, 151335135, 468934515, 3061162125, 10033419375, 68000295825, 234484536825, 1645756410375, 5943863027025, 43105900812975, 162446292283275, 1214871076343925, 4761954230608575
Offset: 3

Views

Author

J. H. Conway

Keywords

Comments

For n >= 0 let A(n) be the product of the positive integers <= n that have the same parity as n minus the product of the positive integers <= n that have the opposite parity as n. Then a(n) = A(n-1) (for n >= 3). [Peter Luschny, Jul 06 2011]

References

  • S. P. Hurd and J. S. McCranie, Quantum factorials. Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994). Congr. Numer. 104 (1994), 19-24.

Links

Crossrefs

Cf. A007912.

Programs

  • Magma
    DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((n-1))-DoubleFactorial(n-2): n in [3..30]]; // Vincenzo Librandi, Aug 08 2017
  • Maple
    DDF := proc(n) local R, P, k; R := {$1..n}; P := select(k->k mod 2 = n mod 2, R); mul(k, k = P) - mul(k, k = R minus P) end: A007911 := n -> DDF(n-1); # Peter Luschny, Jul 06 2011
f:= gfun:-rectoproc({(-n+1)*a(2+n)+a(1+n)+n^2*a(n), a(2)=0,a(3)=1}, a(n), remember):
map(f, [$3..100]); # Robert Israel, Aug 08 2017
  • Mathematica
    Table[(n - 1)!! - (n - 2)!!, {n, 3, 30}] (* Vincenzo Librandi, Aug 08 2017 *)

Formula

(n-1)*a(n+2) = a(n+1) + n^2*a(n). - Robert Israel, Aug 08 2017

A273983 a(n) = ((4*n)!! - (4*n-1)!!)/(4*n+1).

Original entry on oeis.org

1, 31, 2745, 487935, 145769625, 65830256415, 41892106080825, 35736278004165375, 39370290736153001625, 54420772423242699849375, 92234193751998833171261625, 188098544080793843475953349375, 454418941572893462364414856265625, 1283429428883663190972186961851609375
Offset: 1

Views

Author

Chai Wah Wu, Jun 05 2016

Keywords

Comments

Sequence is inspired by A273889. The same argument in A273889 can be used here to prove the expression evaluates to integers.

Links

Crossrefs

Cf. A273889, A007912.

Programs

  • Python
    doublefac=lambda x:1 if x<2 else x*doublefac(x-2)
for i in range(200):
    print(i,(doublefac(4*i)-doublefac(4*i-1))//(4*i+1))
# Brian Cheung, Jun 15 2016

A227415 a(n) = (n+1)!! mod n!!.

Original entry on oeis.org

0, 0, 1, 2, 7, 3, 9, 69, 177, 60, 2715, 4500, 42975, 104580, 91665, 186795, 3493665, 13497435, 97345395, 442245825, 2601636975, 13003053525, 70985324025, 64585694250, 57891366225, 3576632909850, 9411029102475, 147580842959550, 476966861546175, 5708173568847750
Offset: 0

Views

Author

Alex Ratushnyak, Jul 10 2013

Keywords

Comments

a(n) is divisible by A095987(n+1), and is nonzero for n > 1. - Robert Israel, Mar 10 2016

Examples

			a(4) = 5*3 mod 4*2 = 15 mod 8 = 7.

Links

Crossrefs

Cf. A006882, A095987.
Cf. A007911: (n-1)!! - (n-2)!!
Cf. A007912: (n-1)!! - (n-2)!! (mod n).
Cf. A060696: (n-1)!! + (n-2)!! except first two terms.

Programs

  • Maple
    seq(doublefactorial(n+1) mod doublefactorial(n), n=0..100); # Robert Israel, Mar 10 2016
  • Python
    for n in range(2, 77):
    prOdd = prEven = 1
    for i in range(1, n, 2): prOdd *= i
    for i in range(2, n, 2): prEven *= i
    if n&1: print(prEven % prOdd, end=', ')
    else:   print(prOdd % prEven, end=', ')

Formula

a(n) = A006882(n+1) mod A006882(n).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.