A226515 Row 2 of array in A226513.

1, 3, 15, 99, 807, 7803, 87135, 1102419, 15575127, 242943723, 4145495055, 76797289539, 1534762643847, 32907617073243, 753473367606975, 18347287182129459, 473409784213526967, 12902366605394652363, 370357953441110390895, 11167936445234485414179

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))^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

Adjacent sequences: A226512 A226513 A226514 * A226516 A226517 A226518

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 13 2013

Extensions

More terms from Vincenzo Librandi, Jun 18 2013

Status

approved