A087912 Exponential generating function is exp(2*x/(1-x))/(1-x).
1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856, 1815177984, 29808908032, 525586164736, 9898343691264, 198227905206272, 4204989697906688, 94163381359509504, 2219240984918720512, 54898699229094412288, 1422015190821016633344, 38484192401958599131136
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- OEIS Wiki, Exponential generating functions.
Programs
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Magma
m:=25; R
:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 16 2018 -
Maple
a := proc(n) option remember: if n<1 then 1 else (2*n+1)*a(n-1) - (n-1)^2*a(n-2) fi end: 'a(n)'$n=0..17; # Zerinvary Lajos, Sep 26 2006; corrected by M. F. Hasler, Sep 30 2012
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Mathematica
Table[n! SeriesCoefficient[E^(2*x/(1-x))/(1-x), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *) Table[n!*LaguerreL[n, -2], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *) -
PARI
A087912(n)={n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,2^m*x^m/m!^2) ,n)} \\ Paul D. Hanna, Nov 18 2011
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PARI
x='x+O('x^66); Vec(serlaplace(exp(2*x/(1-x))/(1-x))) \\ Joerg Arndt, May 10 2013
Formula
E.g.f.: exp(2*x/(1-x))/(1-x). - M. F. Hasler, Sep 30 2012
a(n) = n!*LaguerreL(n, -2).
a(n) = Sum_{k=0..n} 2^k*(n-k)!*binomial(n, k)^2.
E.g.f.: exp(x) * Sum_{n>=0} 2^n*x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [Paul D. Hanna, Nov 18 2011]
a(n) ~ n^(n+1/4)*exp(2*sqrt(2*n)-n-1)*2^(-3/4). - Vaclav Kotesovec, Sep 29 2012
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(2*n))) = exp(-1). - Vaclav Kotesovec, Oct 12 2016
D-finite with recurrence: a(n) +(-2*n-1)*a(n-1) +(n-1)^2*a(n-2)=0. - R. J. Mathar, Feb 21 2020
Extensions
Several minor edits by M. F. Hasler, Sep 30 2012