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

0,7

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).

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)

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 *)

Prog

(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
(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
(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

Crossrefs

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

Cf. A000579, A173333.

Sequence in context: A280920 A187290 A218487 * A052793 A179728 A052799

Adjacent sequences: A053622 A053623 A053624 * A053626 A053627 A053628

Keyword

easy,nonn

Author

Henry Bottomley, Mar 20 2000

Status

approved