A144297 BINOMIAL transform of A001515.

1, 3, 12, 65, 465, 4212, 46441, 604389, 9071250, 154267865, 2931801639, 61578273462, 1416474723373, 35415138314415, 956276678789100, 27733572777976973, 859779201497486829, 28373745267763162716, 993110842735800666085, 36746019445535955976665

Offset: 0

Author

N. J. A. Sloane, Dec 04 2008

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A001515.

Programs

Magma

I:=[1,3,12]; [n le 3 select I[n] else (2*n-1)*Self(n-1) -(4*n-9)*Self(n-2) +2*(n-3)*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 28 2023

Mathematica

CoefficientList[Series[E^(1+x-Sqrt[1-2*x])/Sqrt[1-2*x],{x,0,20}],x]*Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

SageMath

def A144297_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1+x-sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
A144297_list(40) # G. C. Greubel, Sep 28 2023

Formula

From Vaclav Kotesovec, Oct 20 2012: (Start)

E.g.f.: exp(1+x-sqrt(1-2*x))/sqrt(1-2*x).

Recurrence: a(n) = (2*n+1)*a(n-1) - (4*n-5)*a(n-2) + 2*(n-2)*a(n-3).

a(n) ~ 2^(n+1/2)*n^n/exp(n-3/2). (End)

a(n) = Sum_{j=0..n} binomial(n,j)*A001515(j). - G. C. Greubel, Sep 28 2023