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%I #34 Sep 08 2022 08:45:03
%S 1,7,57,529,5509,63591,805597,11109337,165625929,2654025319,
%T 45481765921,829903882017,16062421776397,328634683136839,
%U 7086337847838789,160604998959958441,3816483607166825617
%N Row sums of unsigned triangle A062138 (generalized a=5 Laguerre).
%H Seiichi Manyama, <a href="/A062192/b062192.txt">Table of n, a(n) for n = 0..441</a> (terms 0..100 from Harry J. Smith)
%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)^6.
%F a(n) = Sum_{m=0..n} n!*binomial(n+5, n-m)/m!.
%F a(n) = (2*n+5)*a(n-1) - (n-1)*(n+4)*a(n-2). - _Vaclav Kotesovec_, Oct 11 2012
%F a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+11/4)/sqrt(2). - _Vaclav Kotesovec_, Oct 11 2012
%F a(n) = n!*LaguerreL(n, 5, -1). - _G. C. Greubel_, Mar 10 2021
%p A062192:= n -> n!*simplify(LaguerreL(n,5,-1), 'LaguerreL');
%p seq(A062192(n), n = 0 .. 30); # _G. C. Greubel_, Mar 10 2021
%t Table[Sum[n! Binomial[n+5,n-m]/m!,{m,0,n}],{n,0,20}] (* _Harvey P. Dale_, Apr 11 2012 *)
%o (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
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1-x))/(1-x)^6)) \\ _G. C. Greubel_, Feb 06 2018
%o (PARI) a(n) = vecsum(apply(abs, Vec(n!*pollaguerre(n, 5)))); \\ _Michel Marcus_, Feb 06 2021
%o (Magma) [Factorial(n)*(&+[Binomial(n+5,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, 5, -1) for n in (0..30)] # _G. C. Greubel_, Mar 10 2021
%Y Cf. A062138, A216294.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jun 19 2001
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