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%I #34 Sep 08 2022 08:46:05
%S 75,308,807,1704,3155,5340,8463,12752,18459,25860,35255,46968,61347,
%T 78764,99615,124320,153323,187092,226119,270920,322035,380028,445487,
%U 519024,601275,692900,794583,907032,1030979,1167180,1316415,1479488,1657227,1850484
%N Column 4 of array in A226513.
%C This is the case h = 4 in Sum_{k=0..h} S2(h,k)*k!*binomial(n+k,k), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [_Bruno Berselli_, Jun 20 2013]
%H Vincenzo Librandi, <a href="/A226741/b226741.txt">Table of n, a(n) for n = 0..1000</a>
%H Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p55">Barred Preferential Arrangements</a>, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (75 - 67*x + 17*x^2 - x^3)/(1 - x)^5.
%F a(n) = (n + 1)^4 + 12*(n + 1)^3 + 36*(n + 1)^2 + 26*(n + 1).
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F E.g.f.: exp(x)*(75 + 233*x + 133*x^2 + 22*x^3 + x^4). - _Franck Maminirina Ramaharo_, Nov 29 2018
%t Table[(n+1)^4 + 12 (n+1)^3 + 36 (n+1)^2 + 26 (n+1), {n, 0, 40}] (* or *) CoefficientList[Series[(75 - 67 x + 17 x^2 - x^3) / (1 - x)^5, {x, 0, 40}], x]
%o (Magma) [(n+1)^4+12*(n+1)^3+36*(n+1)^2+26*(n+1): n in [0..35]] /* or */ I:=[75, 308, 807, 1704, 3155]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];
%Y Cf. columns 2, 3 and 5, 6 of A226513: A005563, A226514, A226800, A226801.
%K nonn,easy
%O 0,1
%A _Vincenzo Librandi_, Jun 18 2013
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