A062192 Row sums of unsigned triangle A062138 (generalized a=5 Laguerre).

1, 7, 57, 529, 5509, 63591, 805597, 11109337, 165625929, 2654025319, 45481765921, 829903882017, 16062421776397, 328634683136839, 7086337847838789, 160604998959958441, 3816483607166825617

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..441 (terms 0..100 from Harry J. Smith)
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A062138, A216294.

Programs

Magma

[Factorial(n)*(&+[Binomial(n+5,n-m)/Factorial(m): m in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Maple

A062192:= n -> n!*simplify(LaguerreL(n,5,-1), 'LaguerreL');
seq(A062192(n), n = 0 .. 30); # G. C. Greubel, Mar 10 2021

Mathematica

Table[Sum[n! Binomial[n+5,n-m]/m!,{m,0,n}],{n,0,20}] (* Harvey P. Dale, Apr 11 2012 *)

PARI

{ f=1; for (n=0, 100, if (n>1, f*=n); a=f*binomial(n+5, n); g=1; a+=sum(m=1, n, f*binomial(n+5, n-m)/g*=m); write("b062192.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 02 2009

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1-x))/(1-x)^6)) \\ G. C. Greubel, Feb 06 2018

PARI

a(n) = vecsum(apply(abs, Vec(n!*pollaguerre(n, 5)))); \\ Michel Marcus, Feb 06 2021

Sage

[factorial(n)*gen_laguerre(n, 5, -1) for n in (0..30)] # G. C. Greubel, Mar 10 2021

Formula

E.g.f.: exp(x/(1-x))/(1-x)^6.

a(n) = Sum_{m=0..n} n!*binomial(n+5, n-m)/m!.

a(n) = (2*n+5)*a(n-1) - (n-1)*(n+4)*a(n-2). - Vaclav Kotesovec, Oct 11 2012

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

a(n) = n!*LaguerreL(n, 5, -1). - G. C. Greubel, Mar 10 2021