A127878 a(n) = n^4 + 4*n^3 + 12*n^2 + 24*n + 24.

24, 65, 168, 393, 824, 1569, 2760, 4553, 7128, 10689, 15464, 21705, 29688, 39713, 52104, 67209, 85400, 107073, 132648, 162569, 197304, 237345, 283208, 335433, 394584, 461249, 536040, 619593, 712568, 815649, 929544, 1054985, 1192728

Offset: 0

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

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A127873-A127883.

Programs

GAP

List([0..40],n->n^4+4*n^3+12*n^2+24*n+24); # Muniru A Asiru, Apr 30 2018

Magma

[n^4 +4*n^3 +12*n^2 +24*n +24: n in [0..30]]; // G. C. Greubel, Apr 29 2018

Maple

seq(n^4+4*n^3+12*n^2+24*n+24,n=0..40); # Muniru A Asiru, Apr 30 2018

Mathematica

Table[24 + 24*n + 12*n^2 + 4*n^3 + n^4, {n, 0, 50}]
LinearRecurrence[{5, -10, 10, -5, 1}, {24, 65, 168, 393, 824}, 50] (* G. C. Greubel, Apr 29 2018 *)

PARI

for(n=0,30, print1(n^4 +4*n^3 +12*n^2 +24*n +24, ", ")) \\ G. C. Greubel, Apr 29 2018

Formula

Integral representation in terms of incomplete Gamma function : a(n)= Exp[n]Gamma[5,n], where Gamma[5,n]= Integrate[x^4 Exp[ -x], {x, n, +infinity}]. - N-E. Fahssi, Jan 25 2008

G.f.: (24 -55*x +83*x^2 -37*x^3 +9*x^4)/(1-x)^5. - Colin Barker, Apr 02 2012

E.g.f.: (24 + 41*x + 31*x^2 + 10*x^3 + x^4)*exp(x). - G. C. Greubel, Apr 29 2018