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The function W_n(6) (see Borwein et al. reference for definition).
4

%I #33 Apr 05 2024 11:07:04

%S 1,20,93,256,545,996,1645,2528,3681,5140,6941,9120,11713,14756,18285,

%T 22336,26945,32148,37981,44480,51681,59620,68333,77856,88225,99476,

%U 111645,124768,138881,154020,170221,187520,205953,225556,246365,268416,291745,316388

%N The function W_n(6) (see Borwein et al. reference for definition).

%H Vincenzo Librandi, <a href="/A169711/b169711.txt">Table of n, a(n) for n = 1..1000</a>

%H Jonathan M. Borwein, Dirk Nuyens, Armin Straub, and James Wan, <a href="https://www.carmamaths.org/resources/jon/walks.pdf">Some Arithmetic Properties of Short Random Walk Integrals</a>, May 2011.

%H Pakawut Jiradilok and Elchanan Mossel, <a href="https://arxiv.org/abs/2402.11990">Gaussian Broadcast on Grids</a>, arXiv:2402.11990 [cs.IT], 2024. See p. 27.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 6*n^3 - 9*n^2 + 4*n. - _Peter Luschny_, May 27 2017

%F G.f.: x*(1+16*x+19*x^2)/(1-x)^4. - _Vincenzo Librandi_, May 28 2017

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, May 28 2017

%p A169711 := proc(n)

%p W(n,6) ;

%p end proc:

%p seq(A169711(n),n=1..20) ; # uses W from A169715; _R. J. Mathar_, Mar 28 2012

%p a := n -> 6*n^3 - 9*n^2 + 4*n: seq(a(n), n=1..33); # _Peter Luschny_, May 27 2017

%t CoefficientList[Series[(1 + 16 x + 19 x^2) / (1 - x)^4, {x, 0, 50}], x] (* or *) Table[6 n^3 - 9 n^2 + 4 n, {n, 1, 40}] (* _Vincenzo Librandi_, May 28 2017 *)

%t LinearRecurrence[{4,-6,4,-1},{1,20,93,256},40] (* _Harvey P. Dale_, Feb 27 2023 *)

%o (Magma) [6*n^3-9*n^2+4*n: n in [1..40]]; // _Vincenzo Librandi_, May 28 2017

%o (PARI) a(n)=6*n^3-9*n^2+4*n \\ _Charles R Greathouse IV_, Oct 18 2022

%Y The sequence in Table 1 of the Borwein et al. reference are A000384, A109711-A109713; A000984, A002893, A002895, A169714, A169715.

%Y Column 3 of A287316.

%Y Cf. A287314.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Apr 17 2010