A269486 - OEIS

%I #20 Feb 16 2025 08:33:30

%S 1,1,683,44287,838861,8138021,51828151,247165843,954437177,3138105961,

%T 9090909091,23775972551,57154490053,128011456717,269971011311,

%U 540609741211,1034834473201,1903994239313,3382547898907,5824512944911

%N a(n) = Sum_{j=0..10} (-n)^j.

%C a(n) = Phi_22(n) where Phi_k(x) is the k-th cyclotomic polynomial.

%H G. C. Greubel, <a href="/A269486/b269486.txt">Table of n, a(n) for n = 0..1000</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Cyclotomic Polynomials at x=n, n! and sigma(n)">Cyclotomic Polynomials at x=n, n! and sigma(n)</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>

%H <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F G.f.: (1 - 10*x + 727*x^2 + 36664*x^3 + 389434*x^4 + 1233508*x^5 + 1365310*x^6 + 534568*x^7 + 66661*x^8 + 1926*x^9 + 11*x^10)/(1-x)^11. - _Vincenzo Librandi_, Feb 29 2016

%F a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).

%F Sum_{n>=0} 1/a(n) = 2.0014880486975...

%t Table[Cyclotomic[22, n], {n, 0, 19}]

%t CoefficientList[Series[(1 -10x +727x^2 +36664x^3 +389434x^4 +1233508x^5 + 1365310 x^6 +534568x^7 +66661x^8 +1926x^9 +11x^10)/(1-x)^11, {x,0,33}], x] (* _Vincenzo Librandi_, Feb 29 2016 *)

%o (PARI) a(n) = polcyclo(22, n); \\ _Michel Marcus_, Feb 28 2016

%o (Magma) [n^10-n^9+n^8-n^7+n^6-n^5+n^4-n^3+n^2-n+1: n in [0..30]] // _Vincenzo Librandi_, Feb 29 2016

%o (Sage) [sum((-n)^j for j in (0..10)) for n in (0..20)] # _G. C. Greubel_, Apr 24 2019

%o (GAP) List([0..20], n-> Sum([0..10], j-> (-n)^j)) # _G. C. Greubel_, Apr 24 2019

%Y Cf. similar sequences of the type Phi_k(n) listed in A269442.

%K nonn,easy,changed

%O 0,3

%A _Ilya Gutkovskiy_, Feb 28 2016

%E Name changed by _G. C. Greubel_, Apr 24 2019