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%I #29 Sep 08 2022 08:45:45
%S 1,2,3,4,6,8,12,24,39,-2,-295,-1308,-3980,-9996,-22150,-44808,-84483,
%T -150534,-256001,-418588,-661806,-1016288,-1521288,-2226376,-3193341,
%U -4498314,-6234123,-8512892,-11468896,-15261684,-20079482,-26142888
%N a(n) = (-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^3 + 22561*n^2 - 7302*n + 2520)/2520.
%C {a(k): 0 <= k < 8} = divisors of 24:
%C a(n) = A027750(A006218(23) + k + 1), 0 <= k < A000005(24).
%H Vincenzo Librandi, <a href="/A161710/b161710.txt">Table of n, a(n) for n = 0..10000</a>
%H Reinhard Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F a(n) = C(n,0) + C(n,1) + C(n,4) - 3*C(n,5) + 8*C(n,6) - 12*C(n,7).
%F G.f.: (1-6*x+15*x^2-20*x^3+16*x^4-12*x^5+18*x^6-24*x^7)/(1-x)^8. - _Bruno Berselli_, Jul 17 2011
%F a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=6, a(5)=8, a(6)=12, a(7)=24, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+ 8*a(n-7)- a(n-8). - _Harvey P. Dale_, Jul 15 2012
%e Differences of divisors of 24 to compute the coefficients of their interpolating polynomial, see formula:
%e 1 ... 2 ... 3 ... 4 ... 6 ... 8 .. 12 .. 24
%e .. 1 ... 1 ... 1 ... 2 ... 2 ... 4 .. 12
%e ..... 0 ... 0 ... 1 ... 0 ... 2 ... 8
%e ........ 0 ... 1 .. -1 ... 2 ... 6
%e ........... 1 .. -2 ... 3 ... 4
%e ............. -3 ... 5 ... 1
%e ................. 8 .. -4
%e .................. -12.
%t Table[(-6n^7+154n^6-1533n^5+7525n^4-18879n^3+22561n^2-7302n+2520)/2520,{n,0,40}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,2,3,4,6,8,12,24},40] (* _Harvey P. Dale_, Jul 15 2012 *)
%o (Magma) [(-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^ 3 + 22561*n^2 - 7302*n + 2520)/2520: n in [0..40]]; // _Vincenzo Librandi_, Jul 17 2011
%o (Python)
%o A161710_list, m = [1], [-12, 80, -223, 333, -281, 127, -23, 1]
%o for _ in range(1,10**2):
%o for i in range(7):
%o m[i+1]+= m[i]
%o A161710_list.append(m[-1]) # _Chai Wah Wu_, Nov 09 2014
%o (PARI) a(n)=(-6*n^7+154*n^6-1533*n^5+7525*n^4-18879*n^3+22561*n^2-7302*n+2520)/2520 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A080856, A161711, A161712, A161713, A161715, A006261, A018253, A161700, A161856.
%K sign,easy
%O 0,2
%A _Reinhard Zumkeller_, Jun 17 2009
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