|
%I #40 Nov 19 2022 10:02:12
%S 1,2,3,5,6,10,15,30,171,886,3359,10143,26072,59502,123931,240048,
%T 438261,761754,1270123,2043641,3188202,4840994,7176951,10416034,
%U 14831391,20758446,28604967,38862163,52116860,69064806,90525155,117456180
%N a(n) = (50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040.
%C {a(k): 0 <= k < 8} = divisors of 30:
%C a(n) = A027750(A006218(29) + k + 1), 0 <= k < A000005(30).
%H Vincenzo Librandi, <a href="/A161715/b161715.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,3) - 3*C(n,4) + 9*C(n,5) - 21*C(n,6) + 50*C(n,7).
%F G.f.: (1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8. - _R. J. Mathar_, Jun 18 2009
%F 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). - _Wesley Ivan Hurt_, Apr 26 2021
%e Differences of divisors of 30 to compute the coefficients of their interpolating polynomial, see formula:
%e 1 2 3 5 6 10 15 30
%e 1 1 2 1 4 5 15
%e 0 1 -1 3 1 10
%e 1 -2 4 -2 9
%e -3 6 -6 11
%e 9 -12 17
%e -21 29
%e 50
%t CoefficientList[Series[(1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8, {x, 0, 50}], x] (* _G. C. Greubel_, Jul 16 2017 *)
%o (Magma) [(50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040: n in [0..40]]; // _Vincenzo Librandi_, Jul 17 2011
%o (Python)
%o A161710_list, m = [1], [50, -321, 864, -1249, 1024, -452, 85, 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) x='x+O('x^50); Vec((1 -6*x +15*x^2 -19*x^3 +8*x^4 +18*x^5 -51*x^6 +84*x^7) /(-1+x)^8) \\ _G. C. Greubel_, Jul 16 2017
%Y Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713.
%Y Cf. A018255, A161700, A161856. - _Reinhard Zumkeller_, Jun 21 2009
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Jun 17 2009
|
