A226738 Row 3 of array in A226513.
1, 4, 24, 184, 1704, 18424, 227304, 3147064, 48278184, 812387704, 14872295784, 294192418744, 6251984167464, 142032703137784, 3434617880825064, 88075274293319224, 2387099326339205544, 68177508876215724664, 2046501717592969431144, 64408432189100396344504
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
- Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Programs
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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
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Mathematica
Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-4, {x, 0, 20}], x]
Formula
E.g.f.: 1/(2 - exp(x))^4 (see the Ahlbach et al. paper, Theorem 4).
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]
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
a(n) ~ n! * n^3 / (96 * log(2)^(n+4)). - Vaclav Kotesovec, Oct 11 2022
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
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * binomial(n,k) * a(n-k).
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)