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%I M2171 N0867 #32 Feb 01 2022 23:30:46
%S 1,-1,-1,2,49,629,6961,38366,-1899687,-133065253,-6482111309,
%T -281940658286,-10702380933551,-247708227641863,14512103549430397,
%U 3377044611825908414,433180638973276282801,47474992085447610990231
%N Related to 3-line Latin rectangles.
%D S. M. Kerawala, The asymptotic number of three-deep Latin rectangles, Bull. Calcutta Math. Soc., 39 (1947), 71-72.
%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 S. M. Kerawala, <a href="/A000186/a000186.pdf">The asymptotic number of three-deep Latin rectangles</a>, Bull. Calcutta Math. Soc., 39 (1947), 71-72. [Annotated scanned copy]
%H S. M. Kerawala, <a href="/A001569/a001569.pdf">Asymptotic solution of the "Probleme des menages</a>, Bull. Calcutta Math. Soc., 39 (1947), 82-84. [Annotated scanned copy]
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%o (Sage)
%o def A001568(N):
%o a = polygen(QQ, 'a')
%o R = PowerSeriesRing(a.parent(), 't', default_prec=N + 2)
%o t = R.gen()
%o n = 1 / t
%o dico = {0: 1}
%o for k in range(1, N + 1):
%o U = sum(di * t**i / factorial(i) for i, di in dico.items())
%o U += a * t**k / factorial(k)
%o U = U.O(k + 2)
%o delta = -U+(n-1)*(n**2-2*n+2)/n**2/(n-2)*U(t=1/(n-1))+(n**2-2*n+2)/n**2/(n-1)*U(t=1/(n-2))+(n**2-2*n-2)/n**2/(n-1)/(n-2)**2*U(t=1/(n-3))+2*(n*n-5*n+3)/n**2/(n-1)/(n-2)**2/(n-3)*U(t=1/(n-4))-4/n**2/(n-2)**2/(n-3)/(n-4)*U(t=1/(n-5))
%o dico[k] = delta[k + 1].numerator().roots()[0][0]
%o return list(dico.values())
%o # _F. Chapoton_, Jan 01 2022
%K sign,more
%O 1,4
%A _N. J. A. Sloane_
%E Signs added by _N. J. A. Sloane_, Jul 23 2015
%E More terms from _F. Chapoton_, Jan 01 2022
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