A062147 - OEIS

%I #38 Jul 31 2024 09:47:24

%S 1,5,31,229,1961,19081,207775,2501801,32989969,472630861,7307593151,

%T 121247816845,2148321709561,40476722545169,807927483311551,

%U 17028146983530961,377844723929464865,8803698102396787861,214877019857456672479,5482159931449737760181

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

%H Vincenzo Librandi, <a href="/A062147/b062147.txt">Table of n, a(n) for n = 0..200</a>

%H Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

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

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

%F a(n) = (2*n+3)*a(n-1) - (n-1)*(n+2)*a(n-2). - _Vaclav Kotesovec_, Oct 11 2012

%F a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+7/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 11 2012

%F a(n) = n!*LaguerreL(n, 3, -1). - _G. C. Greubel_, Mar 10 2021

%p A062147 := n -> n!*simplify(LaguerreL(n,3,-1), 'LaguerreL');

%p seq(A062147(n), n = 0 .. 30); # _G. C. Greubel_, Mar 10 2021

%t Table[Sum[n!*Binomial[n+3,n-k]/k!,{k,0,n}],{n,0,20}]

%t (* or *)

%t Table[n!*SeriesCoefficient[E^(x/(1-x))/(1-x)^4,{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 11 2012 *)

%o (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(x/(1-x))/(1-x)^4)) \\ _Joerg Arndt_, May 06 2013

%o (PARI) a(n) = vecsum(apply(abs,Vec(n!*pollaguerre(n, 3)))); \\ _Michel Marcus_, Feb 06 2021

%o (Magma) [Factorial(n)*(&+[Binomial(n+3,n-m)/Factorial(m): m in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018

%o (Sage) [factorial(n)*gen_laguerre(n, 3, -1) for n in (0..30)] # _G. C. Greubel_, Mar 10 2021

%Y Cf. A062137, A216294.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001