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

0,2

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

Table of n, a(n) for n=0..26.

OEIS Wiki, Eulerian polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

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

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 *)

Prog

(Magma) m:=30; R<x>:=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

Crossrefs

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

Sequence in context: A252978 A268264 A008515 * A000539 A023874 A265839

Adjacent sequences: A179992 A179993 A179994 * A179996 A179997 A179998

Keyword

nonn,easy

Author

Peter Luschny, Aug 05 2010

Status

approved