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A062266 Row sums of unsigned triangle A062140 (generalized a=4 Laguerre).

%I A062266 #39 Jul 11 2025 18:10:20
%S A062266 1,6,43,358,3393,36046,424051,5470158,76751233,1163391958,18941512731,
%T A062266 329604456886,6103575192193,119823200043678,2485452283923043,
%U A062266 54309931242376606,1246803623807490561,29999359707124127398,754865494585690965643,19824604328577866107398
%N A062266 Row sums of unsigned triangle A062140 (generalized a=4 Laguerre).
%H A062266 Vincenzo Librandi, <a href="/A062266/b062266.txt">Table of n, a(n) for n = 0..200</a>
%H A062266 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 A062266 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A062266 E.g.f.: exp(x/(1-x))/(1-x)^5.
%F A062266 a(n) = Sum_{m=0..n} n!*binomial(n+4, n-m)/m!.
%F A062266 a(n) = 2*(n+2)*a(n-1) - (n-1)*(n+3)*a(n-2). - _Vaclav Kotesovec_, Oct 11 2012
%F A062266 a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+9/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 11 2012
%F A062266 a(n) = n!*LaguerreL(n, 4, -1). - _G. C. Greubel_, Mar 10 2021
%p A062266 A062266:= n -> n!*simplify(LaguerreL(n,4,-1), 'LaguerreL');
%p A062266 seq(A062266(n), n = 0 .. 30); # _G. C. Greubel_, Mar 10 2021
%t A062266 Table[n!*SeriesCoefficient[E^(x/(1-x))/(1-x)^5,{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 11 2012 *)
%t A062266 With[{nn=20},CoefficientList[Series[Exp[x/(1-x)]/(1-x)^5,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 11 2025 *)
%o A062266 (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(x/(1-x))/(1-x)^5)) \\ _Joerg Arndt_, May 06 2013
%o A062266 (PARI) a(n) = vecsum(apply(abs,Vec(n!*pollaguerre(n, 4)))); \\ _Michel Marcus_, Feb 06 2021
%o A062266 (Magma) [(&+[Factorial(n)*Binomial(n+4,n-m)/Factorial(m): m in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Feb 06 2018
%o A062266 (Sage) [factorial(n)*gen_laguerre(n, 4, -1) for n in (0..30)] # _G. C. Greubel_, Mar 10 2021
%Y A062266 Cf. A062140, A216294.
%K A062266 nonn,easy
%O A062266 0,2
%A A062266 _Wolfdieter Lang_, Jun 19 2001