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A169712 The function W_n(8) (see Borwein et al. reference for definition).

%I A169712 #31 Apr 05 2024 11:07:04
%S A169712 1,70,639,2716,7885,18306,36715,66424,111321,175870,265111,384660,
%T A169712 540709,740026,989955,1298416,1673905,2125494,2662831,3296140,4036221,
%U A169712 4894450,5882779,7013736,8300425,9756526,11396295,13234564,15286741,17568810,20097331,22889440
%N A169712 The function W_n(8) (see Borwein et al. reference for definition).
%H A169712 Vincenzo Librandi, <a href="/A169712/b169712.txt">Table of n, a(n) for n = 1..1000</a>
%H A169712 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 A169712 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 A169712 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A169712 a(n) = -33*n + 82*n^2 - 72*n^3 + 24*n^4. - _Peter Luschny_, May 27 2017
%F A169712 G.f.: x*(1+65*x+299*x^2+211*x^3)/(1-x)^5. - _Vincenzo Librandi_, May 28 2017
%F A169712 a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - _Vincenzo Librandi_, May 28 2017
%p A169712 A169712 := proc(n)
%p A169712         W(n,8) ;
%p A169712 end proc:
%p A169712 seq(A169712(n),n=1..40) ; # uses W defined in A169715; _R. J. Mathar_, Mar 28 2012
%p A169712 a := n -> -33*n + 82*n^2 - 72*n^3 + 24*n^4:
%p A169712 seq(a(n), n=1..28); # _Peter Luschny_, May 27 2017
%t A169712 Table[-33 n + 82 n^2 - 72 n^3 + 24 n^4, {n, 1, 40}] (* or *) CoefficientList[Series[(1 + 65 x + 299 x^2 + 211 x^3) /(1 - x)^5, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 28 2017 *)
%o A169712 (Magma) [-33*n+82*n^2-72*n^3+24*n^4: n in [1..40]]; // _Vincenzo Librandi_ May 28 2017
%o A169712 (PARI) a(n)=-33*n+82*n^2-72*n^3+24*n^4 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A169712 Column 4 of A287316.
%Y A169712 Cf. A287314.
%K A169712 nonn,easy
%O A169712 1,2
%A A169712 _N. J. A. Sloane_, Apr 17 2010