cp's OEIS Frontend

A169711 The function W_n(6) (see Borwein et al. reference for definition).

%I A169711 #33 Apr 05 2024 11:07:04
%S A169711 1,20,93,256,545,996,1645,2528,3681,5140,6941,9120,11713,14756,18285,
%T A169711 22336,26945,32148,37981,44480,51681,59620,68333,77856,88225,99476,
%U A169711 111645,124768,138881,154020,170221,187520,205953,225556,246365,268416,291745,316388
%N A169711 The function W_n(6) (see Borwein et al. reference for definition).
%H A169711 Vincenzo Librandi, <a href="/A169711/b169711.txt">Table of n, a(n) for n = 1..1000</a>
%H A169711 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 A169711 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 A169711 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A169711 a(n) = 6*n^3 - 9*n^2 + 4*n. - _Peter Luschny_, May 27 2017
%F A169711 G.f.: x*(1+16*x+19*x^2)/(1-x)^4. - _Vincenzo Librandi_, May 28 2017
%F A169711 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, May 28 2017
%p A169711 A169711 := proc(n)
%p A169711         W(n,6) ;
%p A169711 end proc:
%p A169711 seq(A169711(n),n=1..20) ; # uses W from A169715; _R. J. Mathar_, Mar 28 2012
%p A169711 a := n -> 6*n^3 - 9*n^2 + 4*n: seq(a(n), n=1..33); # _Peter Luschny_, May 27 2017
%t A169711 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 A169711 LinearRecurrence[{4,-6,4,-1},{1,20,93,256},40] (* _Harvey P. Dale_, Feb 27 2023 *)
%o A169711 (Magma) [6*n^3-9*n^2+4*n: n in [1..40]]; // _Vincenzo Librandi_, May 28 2017
%o A169711 (PARI) a(n)=6*n^3-9*n^2+4*n \\ _Charles R Greathouse IV_, Oct 18 2022
%Y A169711 The sequence in Table 1 of the Borwein et al. reference are A000384, A109711-A109713; A000984, A002893, A002895, A169714, A169715.
%Y A169711 Column 3 of A287316.
%Y A169711 Cf. A287314.
%K A169711 nonn,easy
%O A169711 1,2
%A A169711 _N. J. A. Sloane_, Apr 17 2010