A179995 Generating function A(5,t)(1+t+t^2)/(1-t)^6, where A(5,t) is an Eulerian polynomial.

1, 33, 276, 1299, 4392, 11925, 27708, 57351, 108624, 191817, 320100, 509883, 781176, 1157949, 1668492, 2345775, 3227808, 4358001, 5785524, 7565667, 9760200, 12437733, 15674076, 19552599, 24164592, 29609625, 35995908

Offset: 0

Author

Peter Luschny, Aug 05 2010

Keywords

• nonn
• easy

Comments

The Eulerian polynomials A(n,t) are here defined in accordance with the Digital Library of Mathematical Functions, Table 26.14.1.

Sums of 3 consecutive fifth powers: a(n) = (n-1)^5+n^5+(n+1)^5. - Bruno Berselli, Jun 24 2013

Links

• OEIS Wiki, Eulerian polynomials.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A158799, A008486, A005918, A027602, A160827 which have generating functions of type A(n, t)(1+t+t^2)/(1-t)^(n+1).

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+27*x+93*x^2+118*x^3+93*x^4+27*x^5+x^6)/(1-x)^6)); // Bruno Berselli, Jun 24 2013

Maple

gfA179995 := proc(t) local i;
add([1,27,93,118,93,27,1][i+1]*t^i,i=0..5)/(1-t)^6 end:
seq(coeff(series(gfA179995(t),t,24),t,j),j=0..16);

Mathematica

Join[{1}, Table[n (3 n^4 + 20 n^2 + 10), {n, 30}]] (* Bruno Berselli, Jun 24 2013 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 33, 276, 1299, 4392, 11925, 27708}, 30] (* Harvey P. Dale, Apr 10 2015 *)

Formula

From Bruno Berselli, Jun 24 2013: (Start)

G.f.: (1 + 27*x + 93*x^2 + 118*x^3 + 93*x^4 + 27*x^5 + x^6) / (1 - x)^6.

a(n) = n*(3*n^4 + 20*n^2 + 10) for n>0, a(0)=1. (End)

a(0)=1, a(1)=33, a(2)=276, a(3)=1299, a(4)=4392, a(5)=11925, a(6)=27708; for n>6, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Apr 10 2015