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%I #29 Jun 05 2017 19:06:58
%S 1,1,3,6,16,43,120,369,1244,4449,16424,61645,233568,890421,3409866,
%T 13105083,50517580,195234557,756198408,2934687173,11408742152,
%U 44420399805,173191793402,676104404123,2642356839108,10337529692357,40481034411830,158658210122079,622329139387184,2442857958597649
%N a(n) = n+binomial(2*n-6,n-3)+binomial(2*n-5,n-3)+binomial(n-1,n-3)+Sum_{i=1..n-3} (binomial(n+i-3,n-3)+2*n-i-5).
%C For n >= 5 this is the number of residuated maps from the lattice N_n to itself.
%H G. C. Greubel, <a href="/A274295/b274295.txt">Table of n, a(n) for n = 0..1000</a>
%H Erika D. Foreman, <a href="http://dx.doi.org/10.18297/etd/2257">Order automorphisms on the lattice of residuated maps of some special nondistributive lattices</a>, (2015). Univ. Louisville, Electronic Theses and Dissertations. Paper 2257.
%F G.f.: -11-12/(x - 1)^3 + x*(-4 + 31/(x-1)^3 + x*(1/sqrt(1 - 4*x) - 23/(x - 1)^3 + x/sqrt(1 - 4*x))). - _Benedict W. J. Irwin_, Aug 09 2016
%F a(n) ~ 5*4^(n-3)/sqrt(Pi*n). - _Ilya Gutkovskiy_, Aug 09 2016
%F Conjecture: (-n+2)*a(n) +(7*n-18)*a(n-1) +14*(-n+3)*a(n-2) +2*(3*n-2)*a(n-3) +(11*n-90)*a(n-4) +(-13*n+102)*a(n-5) +2*(2*n-17)*a(n-6)=0. - _R. J. Mathar_, Oct 07 2016
%p g:=n->n+binomial(2*n-6,n-3)+binomial(2*n-5,n-3)+binomial(n-1,n-3)+add((binomial(n+i-3,n-3)+2*n-i-5),i=1..n-3);
%p [seq(g(n),n=0..40)];
%t Table[n + Binomial[2 * n - 6, n - 3] + Binomial[2 * n - 5, n - 3] + Binomial[n - 1, n - 3] + Sum[(Binomial[n + i - 3, n - 3] + 2 * n - i - 5), {i, 1, n - 3}], {n, 0, 20}] (* _Benedict W. J. Irwin_, Aug 09 2016 *)
%t CoefficientList[Series[-11-12/(x - 1)^3 + x*(-4 + 31/(x-1)^3 + x*(1/Sqrt[1 - 4*x] - 23/(x - 1)^3 + x/Sqrt[1 - 4*x])), {x,0,50}], x] (* _G. C. Greubel_, Jun 05 2017 *)
%o (PARI) x='x+O('x^50); Vec(-11-12/(x - 1)^3 + x*(-4 + 31/(x-1)^3 + x*(1/sqrt(1 - 4*x) - 23/(x - 1)^3 + x/sqrt(1 - 4*x)))) \\ _G. C. Greubel_, Jun 05 2017
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 18 2016
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