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A226738 Row 3 of array in A226513.

%I A226738 #33 Nov 19 2023 08:24:43
%S A226738 1,4,24,184,1704,18424,227304,3147064,48278184,812387704,14872295784,
%T A226738 294192418744,6251984167464,142032703137784,3434617880825064,
%U A226738 88075274293319224,2387099326339205544,68177508876215724664,2046501717592969431144,64408432189100396344504
%N A226738 Row 3 of array in A226513.
%H A226738 Vincenzo Librandi, <a href="/A226738/b226738.txt">Table of n, a(n) for n = 0..100</a>
%H A226738 Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, <a href="http://arxiv.org/abs/1206.6354">Barred Preferential Arrangements</a>, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
%F A226738 E.g.f.: 1/(2 - exp(x))^4 (see the Ahlbach et al. paper, Theorem 4).
%F A226738 a(n) = sum( S2(n,i)*i!*binomial(3+i,i), i=0..n ), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [_Bruno Berselli_, Jun 18 2013]
%F A226738 G.f.: 1/T(0), where T(k) = 1 - x*(k+4)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 28 2013
%F A226738 a(n) ~ n! * n^3 / (96 * log(2)^(n+4)). - _Vaclav Kotesovec_, Oct 11 2022
%F A226738 Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - (n+3)*x/(1 - 2*n*x/(1 - ... ))))))))). - _Peter Bala_, Aug 27 2023
%F A226738 From _Seiichi Manyama_, Nov 19 2023: (Start)
%F A226738 a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * binomial(n,k) * a(n-k).
%F A226738 a(0) = 1; a(n) = 4*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)
%t A226738 Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-4, {x, 0, 20}], x]
%o A226738 (Magma) m:=3; [&+[StirlingSecond(n,i)*Factorial(i)*Binomial(m+i,i): i in [0..n]]: n in [0..20]]; // _Bruno Berselli_, Jun 18 2013
%Y A226738 Cf. rows 0, 1, 2, 4, 5 of A226513: A000670, A005649, A226515, A226739, A226740.
%K A226738 nonn,easy
%O A226738 0,2
%A A226738 _Vincenzo Librandi_, Jun 18 2013