A051431 -id:A051431 - cp's OEIS frontend

A267141 Number of weeks in n! seconds, for n >= 10.

6, 66, 792, 10296, 144144, 2162160, 34594560, 588107520, 10585935360, 201132771840, 4022655436800, 84475764172800, 1858466811801600, 42744736671436800, 1025873680114483200, 25646842002862080000, 666817892074414080000

Offset: 10

Richard H. Sweetman, Jan 10 2016

10! seconds = 6 weeks (exactly)
11! seconds = 66 weeks (exactly)
12! seconds = 792 weeks (exactly), etc.
Demonstration of first term:
  6 w * 7 d/w * 24 h/d * 60 m/h * 60 s/m  = seconds in 6 weeks =
  (2*3)*(1*7)  *(3*8)  * (6*10) * (3*5*4) =
  1*2*3*4*5*6*7*8*(3^2)*10                = 10! seconds

			For n=12, a(12) = 6*(12)!/10! = 792; interpretation: 12! seconds = 792 weeks.

Brady Haran, James Grime, 10!, Numberphile video (2012)

Cf. A051431.

f[n_] := n!/(60*60*24*7); Array[f, 17, 10] (* Robert G. Wilson v, Jan 12 2016 *)

a(n) = 6*n!/10!; \\ Altug Alkan, Jan 12 2016

a(n) = 6*n!/10!.

a(n) = 6*A051431(n-10), for n>=10.

A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 13, 2, 2, 1, 73, 6, 6, 3, 1, 501, 24, 24, 12, 4, 1, 4051, 120, 120, 60, 20, 5, 1, 37633, 720, 720, 360, 120, 30, 6, 1, 394353, 5040, 5040, 2520, 840, 210, 42, 7, 1, 4596553, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 58941091, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
     1;
     1,   1;
     3,   1,   1;
    13,   2,   2,  1;
    73,   6,   6,  3,  1;
   501,  24,  24, 12,  4, 1;
  4051, 120, 120, 60, 20, 5, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 13.

Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431.

Triangle A094587 with 1st column A000262.

T[n_,0]:=n!SeriesCoefficient[Exp[x/(1-x)],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).
T(n,3) = A001710(n-1) for n > 2.
T(n,4) = A001715(n-1) for n > 3.
T(n,5) = A001720(n-1) for n > 4.
T(n,6) = A001725(n-1) for n > 5.
T(n,7) = A001730(n-1) for n > 6.
T(n,8) = A049388(n-8) for n > 7.
T(n,9) = A049389(n-9) for n > 8.
T(n,10) = A049398(n-10) for n > 9.
T(n,11) = A051431(n-11) for n > 10.

cp's OEIS Frontend

A267141 Number of weeks in n! seconds, for n >= 10.

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