A053625 Product of 6 consecutive integers.

0, 0, 0, 0, 0, 0, 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520, 2162160, 3603600, 5765760, 8910720, 13366080, 19535040, 27907200, 39070080, 53721360, 72681840, 96909120, 127512000, 165765600, 213127200, 271252800, 342014400, 427518000, 530122320

Offset: 0

Author

Henry Bottomley, Mar 20 2000

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A002378, A007531, A045619, A052762, A052787.

Cf. A000579, A173333.

Programs

GAP

F:=Factorial;; Concatenation([0,0,0,0,0,0], List([6..30], n-> F(n)/F(n-5) )); # G. C. Greubel, Aug 27 2019

Magma

I:=[0,0,0,0,0,0,720]; [n le 7 select I[n] else 7*Self(n-1) -21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6) +Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012

Maple

seq(combinat[numbperm](n, 6), n=0..31); # Zerinvary Lajos, Apr 26 2007

Mathematica

CoefficientList[Series[720*x^6/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Apr 28 2012 *)
Times@@@Partition[Range[-5,30],6,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,720},30] (* Harvey P. Dale, Nov 13 2015 *)
Pochhammer[Range[30]-6, 6] (* G. C. Greubel, Aug 27 2019 *)

PARI

a(n)=factorback([n-5..n]) \\ Charles R Greathouse IV, Oct 07 2015

Sage

[rising_factorial(n-5,6) for n in (0..30)] # G. C. Greubel, Aug 27 2019

Formula

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5) = n!/(n-6)! = A052787(n)*(n-6) = a(n-1)*n/(n-6).

E.g.f.: x^6*exp(x).

a(n) = 720 * A000579(n). - Zerinvary Lajos, Apr 26 2007

For n > 5: a(n) = A173333(n, n-6). - Reinhard Zumkeller, Feb 19 2010

G.f.: 720*x^6/(1-x)^7. - Colin Barker, Mar 27 2012

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Vincenzo Librandi, Apr 28 2012

From Amiram Eldar, Mar 08 2022: (Start)

Sum_{n>=6} 1/a(n) = 1/600.

Sum_{n>=6} (-1)^n/a(n) = 4*log(2)/15 - 661/3600. (End)