A127877 - OEIS

%I #21 Sep 08 2022 08:45:29

%S 7,115,297,1237,2171,5527,8221,16441,22335,38731,49697,78445,96787,

%T 142927,171381,240817,282551,382051,440665,577861,657387,840775,

%U 945677,1184617,1319791,1624507,1795281,2176861,2388995,2859391,3119077,3691105,4004967,4692307

%N Integers of the form (x^4)/24 + (x^3)/6 + (x^2)/2 + x + 1 with x > 0.

%C Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

%H Colin Barker, <a href="/A127877/b127877.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).

%F From _Colin Barker_, May 15 2016: (Start)

%F a(n) = (11 +5*(-1)^n +16*(2+(-1)^n)*n +18*(3+(-1)^n)*n^2 +36*(1+(-1)^n)*n^3 +54*n^4)/16.

%F a(n) = (27*n^4+36*n^3+36*n^2+24*n+8)/8 for n even.

%F a(n) = (27*n^4+18*n^2+8*n+3)/8 for n odd.

%F a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9) for n>9.

%F G.f.: x*(7+108*x+154*x^2+508*x^3+248*x^4+244*x^5+22*x^6+4*x^7+x^8) / ((1-x)^5*(1+x)^4).

%F (End)

%t a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6 + x^4/24], AppendTo[a, 1 + x + x^2/2 + x^3/6 + x^4/24]], {x, 1, 100}]; a

%t Select[Table[(x^4)/24+(x^3)/6+(x^2)/2+x+1,{x,100}],IntegerQ] (* _Harvey P. Dale_, Aug 14 2012 *)

%o (PARI) Vec(x*(7+108*x+154*x^2+508*x^3+248*x^4+244*x^5+22*x^6+4*x^7+x^8)/((1-x)^5*(1+x)^4) + O(x^50)) \\ _Colin Barker_, May 15 2016

%o (Magma) [(11 +5*(-1)^n +16*(2+(-1)^n)*n +18*(3+(-1)^n)*n^2 +36*(1+(-1)^n)*n^3 +54*n^4)/16: n in [1..30]]; // _G. C. Greubel_, Apr 29 2018

%o (GAP) Filtered(List([0..150],x->(x^4)/24+(x^3)/6+(x^2)/2+x+1),IsInt); # _Muniru A Asiru_, Apr 30 2018

%Y Cf. A127873, A127874, A127875, A127876, A127878, A127879, A127880, A127881, A127882, A127883.

%K nonn,easy

%O 1,1

%A _Artur Jasinski_, Feb 04 2007