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A226801 Column 6 of array in A226513.

%I A226801 #36 Feb 03 2025 18:54:45
%S A226801 4683,25988,87135,227304,507035,1014348,1871583,3242960,5342859,
%T A226801 8444820,12891263,19103928,27595035,38979164,53985855,73472928,
%U A226801 98440523,130045860,169618719,218677640,278946843,352373868,441147935,547719024,674817675,825475508
%N A226801 Column 6 of array in A226513.
%C A226801 This is the case h = 6 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 A226801 Vincenzo Librandi, <a href="/A226801/b226801.txt">Table of n, a(n) for n = 0..1000</a>
%H A226801 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 A226801 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A226801 G.f.: (4683 - 6793*x + 3562*x^2 - 798*x^3 + 67*x^4 - x^5) / (1-x)^7.
%F A226801 a(n) = (n + 1)*(n^5 + 35*n^4 + 430*n^3 + 2320*n^2 + 5525*n + 4683).
%F A226801 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
%F A226801 E.g.f.: exp(x)*(4683 + 21305*x + 19921*x^2 + 6530*x^3 + 890*x^4 + 51*x^5 + x^6). - _Franck Maminirina Ramaharo_, Nov 29 2018
%t A226801 Table[(n + 1) (n^5 + 35 n^4 + 430 n^3 + 2320 n^2 + 5525 n + 4683), {n, 0, 40}] (* or *) CoefficientList[Series[(4683 - 6793 x + 3562 x^2 - 798 x^3 + 67 x^4 - x^5) / (1-x)^7, {x, 0, 30}], x]
%t A226801 LinearRecurrence[{7,-21,35,-35,21,-7,1},{4683,25988,87135,227304,507035,1014348,1871583},30] (* _Harvey P. Dale_, Apr 27 2014 *)
%o A226801 (Magma) [(n+1)*(n^5+35*n^4+430*n^3+2320*n^2+5525*n+4683): n in [0..35]];
%o A226801 (Magma) I:=[4683,25988,87135,227304,507035,1014348,1871583]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..30]];
%Y A226801 Cf. columns 2, 3, 4, 5 of A226513: A005563, A226514, A226741, A226800.
%K A226801 nonn,easy
%O A226801 0,1
%A A226801 _Vincenzo Librandi_, Jun 20 2013