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

0,2

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.

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)

Mathematica

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-4, {x, 0, 20}], x]

Prog

(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

Crossrefs

Cf. rows 0, 1, 2, 4, 5 of A226513: A000670, A005649, A226515, A226739, A226740.

Sequence in context: A152403 A111556 A300736 * A271215 A135905 A239296

Adjacent sequences: A226735 A226736 A226737 * A226739 A226740 A226741

Keyword

nonn,easy

Author

Vincenzo Librandi, Jun 18 2013

Status

approved