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A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

%I A048163 #81 Apr 17 2024 21:22:24
%S A048163 1,2,14,230,6902,329462,22934774,2193664790,276054834902,
%T A048163 44222780245622,8787513806478134,2121181056663291350,
%U A048163 611373265185174628502,207391326125004608457782,81791647413265571604175094,37109390748309009878392597910,19192672725746588045912535407702
%N A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.
%C A048163 a(n) is also the number of max-closed relations on an ordered n-element domain (see the paper by Jeavons and Cooper, 1995). - _Don Knuth_, Feb 12 2024
%D A048163 Lovasz, L. and Vesztergombi, K.; Restricted permutations and Stirling numbers. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 731-738, Colloq. Math. Soc. Janos Bolyai, 18, North-Holland, Amsterdam-New York, 1978.
%D A048163 K. Vesztergombi, Permutations with restriction of middle strength, Stud. Sci. Math. Hungar., 9 (1974), 181-185.
%H A048163 Vincenzo Librandi, <a href="/A048163/b048163.txt">Table of n, a(n) for n = 1..200</a>
%H A048163 Chad Brewbaker, <a href="http://www.emis.de/journals/INTEGERS/papers/i2/i2.Abstract.html">A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues</a>, INTEGERS Vol. 8 (2008), #A02.
%H A048163 Peter G. Jeavons and Martin C. Cooper, <a href="https://doi.org/10.1016/0004-3702(95)00107-7">Tractable constraints on ordered domains</a>, Artificial Intelligence 79 (1995), 327-339.
%H A048163 Hyeong-Kwan Ju and Seunghyun Seo, <a href="http://dx.doi.org/10.1016/j.disc.2012.04.019">Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices</a>, Discrete Math., 312 (2012), 2473-2481.
%H A048163 Ken Kamano, <a href="https://arxiv.org/abs/1701.07157">Lonesum decomposable matrices</a>, arXiv:1701.07157 [math.CO], 2017.
%H A048163 H.-K. Kim et al., <a href="http://arxiv.org/abs/1103.4884">Poly-Bernoulli numbers and lonesum matrices</a>, arXiv:1103.4884 [math.CO], 2011.
%H A048163 Anatol N. Kirillov, <a href="https://doi.org/10.3842/SIGMA.2016.002">On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials</a>, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
%F A048163 E.g.f. (with offset 0): Sum((1-exp(-(m+1)*z))^m, m=0..oo)
%F A048163 O.g.f.: Sum_{n>=1} n^(n-1) * (n-1)! * x^n / Product_{k=1..n-1} (1 - n*k*x). - _Paul D. Hanna_, Jan 05 2013
%F A048163 Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... . - _Vaclav Kotesovec_, Jun 21 2013
%F A048163 a(n) ~ 2*sqrt(Pi) * n^(2*n-3/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n-1)). - _Vaclav Kotesovec_, May 13 2014
%F A048163 a(n+1) = Sum_{k = 0..n} A163626(n,k)^2. - _Philippe Deléham_, May 30 2015
%F A048163 a(n) = A306209(2n-2,n-1). - _Alois P. Heinz_, Feb 01 2019
%F A048163 a(n) = A266695(2n-2). - _Alois P. Heinz_, Apr 17 2024
%e A048163 1
%e A048163 1 + 1 = 2
%e A048163 1 + 9 + 4 = 14
%e A048163 1 + 49 + 144 + 36 = 230
%e A048163 1 + 225 + 2500 + 3600 + 576 = 6902
%e A048163 ... - _Philippe Deléham_, May 30 2015
%t A048163 Table[Sum[((k-1)!)^2*StirlingS2[n,k]^2,{k,1,n}],{n,1,20}] (* _Vaclav Kotesovec_, Jun 21 2013 *)
%o A048163 (PARI) a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1)*(m-1)!*x^m/prod(k=1, m-1, 1+m*k*x+x*O(x^n))), n)) \\ _Paul D. Hanna_, Jan 05 2013
%o A048163 for(n=1,20,print1(a(n),", "))
%o A048163 (PARI) Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
%o A048163 a(n)=sum(k=1,n,(-1)^(n-k)*k^(n-1)*(k-1)!*Stirling2(n-1, k-1))
%o A048163 for(n=1, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jan 06 2013
%o A048163 (PARI) a(n) = sum(k=1, n, (k-1)!^2*stirling(n,k,2)^2); \\ _Michel Marcus_, Jun 22 2018
%Y A048163 Main diagonal of array A099594.
%Y A048163 Cf. A220181, A266695, A306209.
%K A048163 nonn
%O A048163 1,2
%A A048163 _N. J. A. Sloane_, _R. K. Guy_
%E A048163 Entry revised by _N. J. A. Sloane_, Jul 05 2012