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A226515
Row 2 of array in A226513.
14
1, 3, 15, 99, 807, 7803, 87135, 1102419, 15575127, 242943723, 4145495055, 76797289539, 1534762643847, 32907617073243, 753473367606975, 18347287182129459, 473409784213526967, 12902366605394652363, 370357953441110390895, 11167936445234485414179
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))^3 (see the Ahlbach et al. paper, Theorem 4). - Vincenzo Librandi, Jun 18 2013
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(2+i,i), 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/Q(0), where Q(k) = 1 - 3*x*(k + 1) - 2*x^2*(k + 1)*(k + 3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013
G.f.: 1/(1 + x)/Q(0,u), where u = x/(1 + x), Q(k,u) = 1 - u*(3*k + 4) - 2*u^2*(k + 1)*(k + 3)/Q(k+1,u); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^2 /(16*(log(2))^(n + 3)) * (1 + 3*(1 + log(2))/n). - Vaclav Kotesovec, Oct 08 2013
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - (n+2)*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} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*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)^-3, {x, 0, 20}], x] (* Vincenzo Librandi, Jun 18 2013 *)
PROG
(Magma) m:=2; [&+[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, 3, 4, 5 of A226513: A000670, A005649, A226738, A226739, A226740.
Sequence in context: A152402 A361596 A255806 * A135883 A372157 A147664
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 13 2013
EXTENSIONS
More terms from Vincenzo Librandi, Jun 18 2013
STATUS
approved