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A226515 Row 2 of array in A226513.

%I A226515 #45 Nov 19 2023 08:24:50
%S A226515 1,3,15,99,807,7803,87135,1102419,15575127,242943723,4145495055,
%T A226515 76797289539,1534762643847,32907617073243,753473367606975,
%U A226515 18347287182129459,473409784213526967,12902366605394652363,370357953441110390895,11167936445234485414179
%N A226515 Row 2 of array in A226513.
%H A226515 Vincenzo Librandi, <a href="/A226515/b226515.txt">Table of n, a(n) for n = 0..100</a>
%H A226515 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 A226515 E.g.f.: 1/(2 - exp(x))^3 (see the Ahlbach et al. paper, Theorem 4). - _Vincenzo Librandi_, Jun 18 2013
%F A226515 a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(2+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [_Bruno Berselli_, Jun 18 2013]
%F A226515 G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(k + 1) - 2*x^2*(k + 1)*(k + 3)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 02 2013
%F A226515 G.f.: 1/(1 + x)/Q(0,u), where u = x/(1 + x), Q(k,u) = 1 - u*(3*k + 4) - 2*u^2*(k + 1)*(k + 3)/Q(k+1,u); (continued fraction). - _Sergei N. Gladkovskii_, Oct 03 2013
%F A226515 a(n) ~ n! * n^2 /(16*(log(2))^(n + 3)) * (1 + 3*(1 + log(2))/n). - _Vaclav Kotesovec_, Oct 08 2013
%F A226515 Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - (n+2)*x/(1 - 2*n*x/(1 - ... ))))))))). - _Peter Bala_, Aug 27 2023
%F A226515 From _Seiichi Manyama_, Nov 19 2023: (Start)
%F A226515 a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
%F A226515 a(0) = 1; a(n) = 3*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)
%t A226515 Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-3, {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 18 2013 *)
%o A226515 (Magma) m:=2; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]]; // _Bruno Berselli_, Jun 18 2013
%Y A226515 Cf. rows 0, 1, 3, 4, 5 of A226513: A000670, A005649, A226738, A226739, A226740.
%K A226515 nonn,easy
%O A226515 0,2
%A A226515 _N. J. A. Sloane_, Jun 13 2013
%E A226515 More terms from _Vincenzo Librandi_, Jun 18 2013