A070070 - OEIS

%I #27 Feb 16 2025 08:32:46

%S 0,2,7,17,35,64,111,182,286,436,647,938,1336,1871,2583,3520,4741,6320,

%T 8347,10930,14199,18312,23460,29869,37808,47600,59624,74331,92250,

%U 114006,140329,172077,210249,256010,310717,375943,453513,545538,654453

%N Rounded value of n*L_n(-1) where L is the Laguerre polynomial.

%H G. C. Greubel, <a href="/A070070/b070070.txt">Table of n, a(n) for n = 0..450</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial.</a>

%F a(n) ~ exp(2*sqrt(n) - 1/2) * n^(3/4) / (2*sqrt(Pi)) * (1 + 31/(48*sqrt(n))). - _Vaclav Kotesovec_, Nov 14 2017

%F a(n) = round(n * [x^n] exp(x/(1 - x))/(1 - x)). - _Ilya Gutkovskiy_, Jun 05 2018

%F a(n) = round(n * Sum_{k=0..n} binomial(n, k) / k!). - _Sean A. Irvine_, Jun 01 2024

%p a := n->round(expand(n*LaguerreL(n,-1)));

%t a[n_] := Round[n*LaguerreL[n, 0, -1]]

%o (PARI) {a(n) = if( n<0, 0, round(n * polcoeff(exp(x/(1-x) + x*O(x^n)) / (1-x), n)))}; /* _Michael Somos_, Dec 04 2002 */

%o (PARI) for(n=0,40, print1(round(n*sum(k=0,n, binomial(n,k)/k!)), ", ")) \\ _G. C. Greubel_, May 14 2018

%o (Magma) [Round(n*(&+[Binomial(n,k)/Factorial(k): k in [0..n]])): n in [0..40]]; // _G. C. Greubel_, May 14 2018

%K nonn,changed

%O 0,2

%A _Karol A. Penson_, Apr 19 2002