cp's OEIS Frontend

A226800 Column 5 of array in A226513.

%I A226800 #43 Feb 03 2025 18:54:54
%S A226800 541,2612,7803,18424,37625,69516,119287,193328,299349,446500,645491,
%T A226800 908712,1250353,1686524,2235375,2917216,3754637,4772628,5998699,
%U A226800 7463000,9198441,11240812,13628903,16404624,19613125,23302916,27525987,32337928,37798049,43969500
%N A226800 Column 5 of array in A226513.
%C A226800 This is the case h = 5 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 A226800 Vincenzo Librandi, <a href="/A226800/b226800.txt">Table of n, a(n) for n = 0..1000</a>
%H A226800 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 A226800 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A226800 G.f.: (541 - 634*x + 246*x^2 - 34*x^3 + x^4)/(1 - x)^6.
%F A226800 a(n) = (n + 1)*(n^4 + 24*n^3 + 186*n^2 + 554*n + 541).
%F A226800 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
%F A226800 E.g.f.: exp(x)*(541 + 2071*x + 1560*x^2 + 385*x^3 + 35*x^4 + x^5). - _Franck Maminirina Ramaharo_, Nov 29 2018
%t A226800 Table[(n + 1)^5 + 20 (n+1)^4 + 120 (n+1)^3 + 250 (n+1)^2 + 150 (n+1), {n, 0, 40}] (* or *) CoefficientList[Series[(541 - 634 x + 246 x^2 - 34 x^3 + x^4) / (1 - x)^6, {x, 0, 30}], x]
%o A226800 (Magma) [(n+1)^5+20*(n+1)^4+120*(n+1)^3+250*(n+1)^2+150*(n+1): n in [0..30]];
%o A226800 (Magma) I:=[541,2612,7803,18424,37625,69516]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..30]];
%Y A226800 Cf. columns 2, 3, 4 and 6 of A226513: A005563, A226514, A226741, A226801.
%K A226800 nonn,easy
%O A226800 0,1
%A A226800 _Vincenzo Librandi_, Jun 18 2013