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%I M5185 N2253 #42 May 02 2022 08:31:38
%S 1,25,450,7350,117600,1905120,31752000,548856000,9879408000,
%T 185513328000,3636061228800,74373979680000,1586644899840000,
%U 35272336619520000,816302647480320000,19645683716026368000,491142092900659200000,12740803704070041600000
%N Coefficients of Laguerre polynomials.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
%D Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001811/b001811.txt">Table of n, a(n) for n = 4..100</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Cornelius Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a>. (Annotated scans of selected pages)
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>.
%F a(n) = n!*n*(n-1)(n-2)(n-3)/(4!)^2. a(4)=1, a(n+1) = a(n) * (n+1)^2 / (n-3).
%F a(n) = A021009(n, 4), n >= 4.
%F E.g.f.: x^4/(4!*(1-x)^5).
%F If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n) = (-1)^n*f(n,4,-5), (n >= 4). - _Milan Janjic_, Mar 01 2009
%F From _Amiram Eldar_, May 02 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 64*(Ei(1) - gamma - e) + 272/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
%F Sum_{n>=4} (-1)^n/a(n) = 544*(gamma - Ei(-1)) - 320/e - 944/3, where Ei(-1) = -A099285. (End)
%e G.f. = x^4 + 25*x^5 + 450*x^6 + 7350*x^7 + 117600*x^8 + 1905120*x^9 + ...
%p with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card<r), U=Sequence(Z, card>=1)}, labeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..19) ; # _Zerinvary Lajos_, Feb 07 2008
%t Table[n! n (n - 1) (n - 2) (n - 3)/(4!)^2, {n, 4, 20}] (* _T. D. Noe_, Aug 10 2012 *)
%o (Sage) [factorial(m) * binomial(m, 4) / 24 for m in range(4,19)] # _Zerinvary Lajos_, Jul 05 2008
%Y Cf. A053495.
%Y Cf. A001113, A001620, A091725, A099285.
%K nonn,easy
%O 4,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Feb 07 2001
%E Corrected by _T. D. Noe_, Aug 10 2012
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