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A226514 Column 3 of array in A226513.

%I A226514 #24 Sep 08 2022 08:46:05
%S A226514 13,44,99,184,305,468,679,944,1269,1660,2123,2664,3289,4004,4815,5728,
%T A226514 6749,7884,9139,10520,12033,13684,15479,17424,19525,21788,24219,26824,
%U A226514 29609,32580,35743,39104,42669,46444,50435,54648,59089,63764,68679,73840,79253,84924,90859,97064,103545,110308,117359,124704
%N A226514 Column 3 of array in A226513.
%C A226514 This is the case h=3 in sum( S2(h,k)*k!*binomial(n+k,k), k=0..h ), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [_Bruno Berselli_, Jun 20 2013]
%H A226514 Vincenzo Librandi, <a href="/A226514/b226514.txt">Table of n, a(n) for n = 0..1000</a>
%H A226514 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 A226514 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A226514 a(n) = (n + 1)*(n^2 + 8*n + 13).
%F A226514 G.f.: (13 - 8*x + x^2)/(1 - x)^4. - _Vincenzo Librandi_, Jun 17 2013
%F A226514 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 17 2013
%F A226514 E.g.f.: exp(x)*(13 + 31*x + 12*x^2 + x^3). - _Franck Maminirina Ramaharo_, Nov 29 2018
%t A226514 Table[n^3 + 9*n^2 + 21*n + 13, {n, 0, 50}] (* or *) CoefficientList[Series[(13 - 8 x + x^2) / (1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 17 2013 *)
%o A226514 (Magma) [n^3+9*n^2+21*n+13: n in [0..50]]; // _Vincenzo Librandi_, Jun 17 2013
%o A226514 (Magma) I:=[13, 44, 99, 184]; [n le 4 select I[n] else 4*Self(n-1) -6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 17 2013
%Y A226514 Cf. columns 2 and 4, 5, 6 of A226513: A005563, A226741, A226800, A226801.
%K A226514 nonn,easy
%O A226514 0,1
%A A226514 _N. J. A. Sloane_, Jun 13 2013