A062147 Row sums of unsigned triangle A062137 (generalized a=3 Laguerre).

1, 5, 31, 229, 1961, 19081, 207775, 2501801, 32989969, 472630861, 7307593151, 121247816845, 2148321709561, 40476722545169, 807927483311551, 17028146983530961, 377844723929464865, 8803698102396787861, 214877019857456672479, 5482159931449737760181

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A062137, A216294.

Programs

Magma

[Factorial(n)*(&+[Binomial(n+3,n-m)/Factorial(m): m in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Maple

A062147 := n -> n!*simplify(LaguerreL(n,3,-1), 'LaguerreL');
seq(A062147(n), n = 0 .. 30); # G. C. Greubel, Mar 10 2021

Mathematica

Table[Sum[n!*Binomial[n+3,n-k]/k!,{k,0,n}],{n,0,20}]
(* or *)
Table[n!*SeriesCoefficient[E^(x/(1-x))/(1-x)^4,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 11 2012 *)

PARI

my(x='x+O('x^66)); Vec(serlaplace(exp(x/(1-x))/(1-x)^4)) \\ Joerg Arndt, May 06 2013

PARI

a(n) = vecsum(apply(abs,Vec(n!*pollaguerre(n, 3)))); \\ Michel Marcus, Feb 06 2021

Sage

[factorial(n)*gen_laguerre(n, 3, -1) for n in (0..30)] # G. C. Greubel, Mar 10 2021

Formula

E.g.f.: exp(x/(1-x))/(1-x)^4.

a(n) = Sum_{m=0..n} n!*binomial(n+3, n-m)/m!.

a(n) = (2*n+3)*a(n-1) - (n-1)*(n+2)*a(n-2). - Vaclav Kotesovec, Oct 11 2012

a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+7/4)/sqrt(2). - Vaclav Kotesovec, Oct 11 2012

a(n) = n!*LaguerreL(n, 3, -1). - G. C. Greubel, Mar 10 2021