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A053625 Product of 6 consecutive integers. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 10:20 EST 2022. Contains 358586 sequences. (Running on oeis4.)