A277373 - OEIS

%I #57 Feb 16 2025 08:33:36

%S 1,2,14,168,2840,61870,1649232,51988748,1891712384,78031713690,

%T 3598075308800,183396819358192,10239159335648256,621414669926828102,

%U 40733145577028065280,2867932866586451980500,215859025837098699948032,17295664826665032427023922,1469838791737283957748596736

%N a(n) = Sum_{k=0..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.

%C Limit_{n -> infinity} (LaguerreL(n,-n)/BesselI(0,2*n))^(1/n) = exp(-2 + 1/phi) * phi^2 = 0.657347578792874..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 12 2016

%C For m > 0, n!*LaguerreL(n, -m*n) ~ sqrt(1/2 + (m+2)/(2*sqrt(m*(m+4)))) * (2+m+sqrt(m*(m+4)))^n * exp(n*(sqrt(m*(m+4))-m-2)/2) * n^n / 2^n. - _Vaclav Kotesovec_, Oct 14 2016

%C For m > 4, (-1)^n * n! * LaguerreL(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n. - _Vaclav Kotesovec_, Feb 20 2020

%H Alois P. Heinz, <a href="/A277373/b277373.txt">Table of n, a(n) for n = 0..356</a>

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">Modified Bessel Function of the First Kind</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

%F a(n) = p(n,n) where p(n,x) = Sum_{k=0..n} binomial(n,n-k)*x^(n-k)*n!/(n-k)!. The coefficients of these polynomials are in A144084 (sorted by falling powers).

%F a(n) = n!*LaguerreL(n, -n).

%F a(n) = (-1)^n*KummerU(-n, 1, -n).

%F a(n) = n^n*hypergeom([-n, -n], [], 1/n) for n>=1.

%F a(n) ~ n^n * phi^(2*n+1) * exp(n/phi-n) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 12 2016

%F a(n) = n! * [x^n] exp(n*x/(1-x))/(1-x). - _Alois P. Heinz_, Jun 28 2017

%F a(n) = n!^2 * [x^n] exp(x) * BesselI(0,2*sqrt(n*x)). - _Ilya Gutkovskiy_, Jun 19 2022

%p A277373 := n -> n!*LaguerreL(n, -n): seq(simplify(A277373(n)), n=0..18);

%p # second Maple program:

%p a:= n-> n! * add(binomial(n, i)*n^i/i!, i=0..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 27 2017

%t Table[n!*LaguerreL[n, -n], {n, 0, 30}] (* _G. C. Greubel_, May 16 2018 *)

%o (Sage)

%o @cached_function

%o def L(n, x):

%o     if n == 0: return 1

%o     if n == 1: return 1 - x

%o     return (L(n-1,x) * (2*n-1-x) - L(n-2,x)*(n-1))/n

%o A277373 = lambda n: factorial(n)*L(n, -n)

%o print([A277373(n) for n in (0..20)])

%o (PARI) a(n) = sum(k=0,n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!) \\ _Charles R Greathouse IV_, Feb 07 2017

%o (PARI) a(n) = n!*pollaguerre(n, 0, -n); \\ _Michel Marcus_, Feb 05 2021

%o (Magma) [(&+[Binomial(n, n-k)*Binomial(n, k)*n^(n-k)*Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 16 2018

%Y Cf. A002720 (n!L(n,-1)), A087912 (n!L(n,-2)), A277382 (n!L(n,-3)), A277372 (n!L(n,-n)-n^n), A277423 (n!L(n,n)), A144084 (polynomials).

%Y Cf. A277391 (n!L(n,-2*n)), A277392 (n!L(n,-3*n)), A277418 (n!L(n,-4*n)), A277419 (n!L(n,-5*n)), A277420 (n!L(n,-6*n)), A277421 (n!L(n,-7*n)), A277422 (n!L(n,-8*n)).

%Y Main diagonal of A289192.

%K nonn,nice

%O 0,2

%A _Peter Luschny_, Oct 12 2016