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URL: https://oeis.org/A053625

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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
OFFSET
0,7
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
(SageMath) [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
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Mar 20 2000
STATUS
approved