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Row sums of unsigned triangle A062140 (generalized a=4 Laguerre).
5

%I #37 Jul 31 2024 09:47:33

%S 1,6,43,358,3393,36046,424051,5470158,76751233,1163391958,18941512731,

%T 329604456886,6103575192193,119823200043678,2485452283923043,

%U 54309931242376606,1246803623807490561,29999359707124127398,754865494585690965643,19824604328577866107398

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

%H Vincenzo Librandi, <a href="/A062266/b062266.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)^5.

%F a(n) = Sum_{m=0..n} n!*binomial(n+4, n-m)/m!.

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

%F a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+9/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 11 2012

%F a(n) = n!*LaguerreL(n, 4, -1). - _G. C. Greubel_, Mar 10 2021

%p A062266:= n -> n!*simplify(LaguerreL(n,4,-1), 'LaguerreL');

%p seq(A062266(n), n = 0 .. 30); # _G. C. Greubel_, Mar 10 2021

%t Table[n!*SeriesCoefficient[E^(x/(1-x))/(1-x)^5,{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)^5)) \\ _Joerg Arndt_, May 06 2013

%o (PARI) a(n) = vecsum(apply(abs,Vec(n!*pollaguerre(n, 4)))); \\ _Michel Marcus_, Feb 06 2021

%o (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 (Sage) [factorial(n)*gen_laguerre(n, 4, -1) for n in (0..30)] # _G. C. Greubel_, Mar 10 2021

%Y Cf. A062140, A216294.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001