A059115 Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).

1, 2, 9, 58, 485, 4986, 60877, 861554, 13878153, 250854130, 5030058161, 110837000682, 2662669300909, 69270266115818, 1940260799150325, 58220372514830626, 1863293173842259217, 63356877145370671074

Offset: 0

Author

Vladeta Jovovic, Jan 06 2001

Keywords

• easy
• nonn

Comments

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

Examples

			(1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A001861, A049020, A052897, A059099, A059110.

Row sums of A059114.

Programs

Magma

[Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

Maple

s := series((1-x)/(1-2*x)*exp(x/(1-x)), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2020 *)
Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

PARI

{a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* Michael Somos, Aug 03 2006 */

PARI

{a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* Michael Somos, Aug 03 2006 */

PARI

a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ Michel Marcus, Feb 23 2021

Sage

[factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

Formula

Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).

Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - Michael Somos, Aug 03 2006

a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - G. C. Greubel, Feb 23 2021

a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Feb 23 2021

Extensions

Definition clarified by Harvey P. Dale, Jul 18 2020