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A226739
Row 4 of array in A226513.
5
1, 5, 35, 305, 3155, 37625, 507035, 7608305, 125687555, 2265230825, 44210200235, 928594230305, 20880079975955, 500343586672025, 12726718227077435, 342425052939060305, 9715738272696568355, 289901469137229041225, 9074304882434034258635, 297297854264669632338305
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))^5 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(4+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k + 1 + m) - 2*x^2*(k + 1)*(k + 1 + m)/Q(k+1), m = 4 is row 4 of array in A226513; (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^4 / (768 * log(2)^(n+5)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - (n+4)*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} (4*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5*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)^-5, {x, 0, 20}], x]
PROG
(Magma) m:=4; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];
CROSSREFS
Cf. rows 0, 1, 2, 3 and 5 of A226513: A000670, A005649, A226515, A226738, A226740.
Sequence in context: A177354 A253096 A305964 * A109253 A052797 A371540
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Jun 18 2013
STATUS
approved