A166474 a(1)=1; a(2)=2; for n>2, a(n) = a(n-1) + A000217(n-1)*a(n-2).

1, 2, 5, 17, 67, 322, 1729, 10745, 72989, 556514, 4570909, 41300833, 397831735, 4156207538, 45928539713, 544673444273, 6790954845241, 90125991819010, 1251379270355221, 18375317715967121, 281164964490563531, 4525863356878968482

Offset: 1

Author

Matthew Vandermast, Nov 05 2009

Keywords

• nonn

Comments

Equals the eigensequence of an infinite lower triangular matrix with 1's in the main diagonal and the triangular series in the subdiagonal.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A066120, A166469, A351046.

Programs

Haskell

a166474 n = a166474_list !! (n-1)
a166474_list = 1 : 2 : zipWith (+)
   (tail a166474_list) (zipWith (*) a166474_list $ drop 2 a000217_list)
-- Reinhard Zumkeller, Feb 27 2012

Magma

[n le 2 select n else Self(n-1) + Binomial(n,2)*Self(n-2): n in [1..41]]; // G. C. Greubel, Aug 02 2024

Mathematica

Rest[CoefficientList[Series[-2*E^(Sqrt[2]*ArcTanh[x/Sqrt[2]])/(x^2-2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 19 2012 *)

SageMath

@CachedFunction
def A166474(n):
    if n<3: return n
    else: return A166474(n-1) + binomial(n,2)*A166474(n-2)
[A166474(n) for n in range(1,41)] # G. C. Greubel, Aug 02 2024

Formula

a(n+1) = A166469(A066120(n)).

E.g.f.: -2*exp(sqrt(2)*arctanh(x/sqrt(2)))/(x^2-2) = ((sqrt(2) + x)^2/(2 - x^2))^(1/sqrt(2))*2/(2 - x^2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ n!*2^(1/sqrt(2)-n/2)*n^(1/sqrt(2))/(2*Gamma(1+1/sqrt(2))). - Vaclav Kotesovec, Oct 19 2012

Extensions

More terms from Sean A. Irvine, Jun 16 2011