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A277373 a(n) = Sum_{k=0..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!. 20
1, 2, 14, 168, 2840, 61870, 1649232, 51988748, 1891712384, 78031713690, 3598075308800, 183396819358192, 10239159335648256, 621414669926828102, 40733145577028065280, 2867932866586451980500, 215859025837098699948032, 17295664826665032427023922, 1469838791737283957748596736 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Lim 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

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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..356

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind

Wikipedia, Laguerre polynomials

FORMULA

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).

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

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

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

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

a(n) = n! * [x^n] exp(n*x/(1-x))/(1-x). - Alois P. Heinz, Jun 28 2017

MAPLE

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

# second Maple program:

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

seq(a(n), n=0..20);  # Alois P. Heinz, Jun 27 2017

MATHEMATICA

Table[n!*LaguerreL[n, -n], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)

PROG

(Sage)

@cached_function

def L(n, x):

    if n == 0: return 1

    if n == 1: return 1 - x

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

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

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

(PARI) a(n) = sum(k=0, n, binomial(n, n-k)*n^(n-k)*n!/(n-k)!) \\ Charles R Greathouse IV, Feb 07 2017

(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

CROSSREFS

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).

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)).

Main diagonal of A289192.

Sequence in context: A245896 A124215 A003582 * A084946 A047055 A229257

Adjacent sequences:  A277370 A277371 A277372 * A277374 A277375 A277376

KEYWORD

nonn,nice

AUTHOR

Peter Luschny, Oct 12 2016

STATUS

approved

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Last modified March 30 18:37 EDT 2020. Contains 333127 sequences. (Running on oeis4.)