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A226739 Row 4 of array in A226513.

%I A226739 #42 Nov 19 2023 08:24:40
%S A226739 1,5,35,305,3155,37625,507035,7608305,125687555,2265230825,
%T A226739 44210200235,928594230305,20880079975955,500343586672025,
%U A226739 12726718227077435,342425052939060305,9715738272696568355,289901469137229041225,9074304882434034258635,297297854264669632338305
%N A226739 Row 4 of array in A226513.
%H A226739 Vincenzo Librandi, <a href="/A226739/b226739.txt">Table of n, a(n) for n = 0..100</a>
%H A226739 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.
%F A226739 E.g.f.: 1/(2 - exp(x))^5 (see the Ahlbach et al. paper, Theorem 4).
%F A226739 a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(4+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
%F A226739 G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k + 1 + m) - 2*x^2*(k + 1)*(k + 1 + m)/Q(k+1), m = 4 is row 4 of array in A226513; (continued fraction). - _Sergei N. Gladkovskii_, Oct 03 2013
%F A226739 a(n) ~ n! * n^4 / (768 * log(2)^(n+5)). - _Vaclav Kotesovec_, Oct 11 2022
%F A226739 Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - (n+4)*x/(1 - 2*n*x/(1 - ... ))))))))). - _Peter Bala_, Aug 27 2023
%F A226739 From _Seiichi Manyama_, Nov 19 2023: (Start)
%F A226739 a(0) = 1; a(n) = Sum_{k=1..n} (4*k/n + 1) * binomial(n,k) * a(n-k).
%F A226739 a(0) = 1; a(n) = 5*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)
%t A226739 Range[0,20]! CoefficientList[Series[(2 - Exp@x)^-5, {x, 0, 20}],x]
%o A226739 (Magma) m:=4; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];
%Y A226739 Cf. rows 0, 1, 2, 3 and 5 of A226513: A000670, A005649, A226515, A226738, A226740.
%K A226739 nonn,easy
%O A226739 0,2
%A A226739 _Vincenzo Librandi_, Jun 18 2013