A226740 Row 5 of array in A226513.

1, 6, 48, 468, 5340, 69516, 1014348, 16372908, 289366860, 5553635436, 114964523148, 2552305112748, 60474398655180, 1522843616043756, 40605864407444748, 1142786353739186988, 33848016050071188300, 1052381222812017946476, 34266937867683980363148, 1166071764343727862515628

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))^6 (see the Ahlbach et al. paper, Theorem 4).

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).

a(n) ~ n! * n^5 / (7680 * log(2)^(n+6)). - Vaclav Kotesovec, Oct 11 2022

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

From Seiichi Manyama, Nov 19 2023: (Start)

a(0) = 1; a(n) = Sum_{k=1..n} (5*k/n + 1) * binomial(n,k) * a(n-k).

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)

Mathematica

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-6, {x, 0, 20}], x]

Prog

(Magma) m:=5; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Crossrefs

Cf. rows 0, 1, 2, 3, 4 of A226513: A000670, A005649, A226515, A226738, A226739.

Sequence in context: A365187 A319292 A261834 * A244509 A105627 A051578

Adjacent sequences: A226737 A226738 A226739 * A226741 A226742 A226743

Keyword

nonn,easy

Author

Vincenzo Librandi, Jun 18 2013

Status

approved