A127877 Integers of the form (x^4)/24 + (x^3)/6 + (x^2)/2 + x + 1 with x > 0.

7, 115, 297, 1237, 2171, 5527, 8221, 16441, 22335, 38731, 49697, 78445, 96787, 142927, 171381, 240817, 282551, 382051, 440665, 577861, 657387, 840775, 945677, 1184617, 1319791, 1624507, 1795281, 2176861, 2388995, 2859391, 3119077, 3691105, 4004967, 4692307

Offset: 1

Author

Artur Jasinski, Feb 04 2007

Keywords

• nonn
• easy

Comments

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).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Crossrefs

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

Programs

GAP

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

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

Mathematica

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

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

Formula

From Colin Barker, May 15 2016: (Start)

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.

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

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

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.

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).

(End)