login
A226740
Row 5 of array in A226513.
5
1, 6, 48, 468, 5340, 69516, 1014348, 16372908, 289366860, 5553635436, 114964523148, 2552305112748, 60474398655180, 1522843616043756, 40605864407444748, 1142786353739186988, 33848016050071188300, 1052381222812017946476, 34266937867683980363148, 1166071764343727862515628
OFFSET
0,2
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Jun 18 2013
STATUS
approved