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A226740 Row 5 of array in A226513.

%I A226740 #30 Nov 19 2023 08:24:35
%S A226740 1,6,48,468,5340,69516,1014348,16372908,289366860,5553635436,
%T A226740 114964523148,2552305112748,60474398655180,1522843616043756,
%U A226740 40605864407444748,1142786353739186988,33848016050071188300,1052381222812017946476,34266937867683980363148,1166071764343727862515628
%N A226740 Row 5 of array in A226513.
%H A226740 Vincenzo Librandi, <a href="/A226740/b226740.txt">Table of n, a(n) for n = 0..100</a>
%H A226740 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 A226740 E.g.f.: 1/(2 - exp(x))^6 (see the Ahlbach et al. paper, Theorem 4).
%F A226740 a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(5+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
%F A226740 a(n) ~ n! * n^5 / (7680 * log(2)^(n+6)). - _Vaclav Kotesovec_, Oct 11 2022
%F A226740 Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 8*x/(1 - 6*x/(1 - (n+5)*x/(1 - 2*n*x/(1 - ... ))))))))). - _Peter Bala_, Aug 27 2023
%F A226740 From _Seiichi Manyama_, Nov 19 2023: (Start)
%F A226740 a(0) = 1; a(n) = Sum_{k=1..n} (5*k/n + 1) * binomial(n,k) * a(n-k).
%F A226740 a(0) = 1; a(n) = 6*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)
%t A226740 Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-6, {x, 0, 20}], x]
%o A226740 (Magma) m:=5; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];
%Y A226740 Cf. rows 0, 1, 2, 3, 4 of A226513: A000670, A005649, A226515, A226738, A226739.
%K A226740 nonn,easy
%O A226740 0,2
%A A226740 _Vincenzo Librandi_, Jun 18 2013