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%I #85 May 13 2023 18:08:20
%S 1,1,4,18,88,450,2364,12642,68464,374274,2060980,11414898,63521352,
%T 354870594,1989102444,11180805570,63001648608,355761664002,
%U 2012724468324,11406058224594,64734486343480,367891005738690,2093292414443164,11923933134635298,67990160422313808
%N a(n) = T(n,n), array T as in A050143.
%C Also main diagonal of array : m(i,1)=1, i>=1; m(1,j)=2, j>1; m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 2 2 ... / 1 4 8 12 ... / 1 6 18 38 ... / 1 8 32 88 ... / - _Benoit Cloitre_, Aug 05 2002
%C a(n) is also the number of order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - _Abdullahi Umar_, Aug 25 2008
%C Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0<=r<=n, and the other terms recursively with T(r,c) = T(r,c-1) + 2*T(r-1,c-1). The sum of the last terms in each row is Sum_{r=0..n} T(r,r)=a(n+1). For n=4 the triangle is 1; 4 6; 6 14 26; 4 16 44 96; 1 9 41 129 321 with the sum of the last terms being 1 + 6 + 26 + 96 + 321 = 450 = a(5). - _J. M. Bergot_, Jan 29 2013
%C It may be better to define a(0) = 0 for formulas without exceptions. - _Michael Somos_, Nov 25 2016
%C a(n) is the number of points at L1 distance n-1 from any point in Z^n, for n>=1. - _Shel Kaphan_, Mar 24 2023
%H Vincenzo Librandi, <a href="/A050146/b050146.txt">Table of n, a(n) for n = 0..200</a>
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">A. Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra, 278 (2004), 342-359.
%H A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq., 7 (2004), 04.3.8.
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 4.
%H Emanuele Munarini, <a href="http://www.emis.de/journals/INTEGERS/papers/j29/j29.Abstract.html">Combinatorial properties of the antichains of a garland</a>, Integers, 9 (2009), 353-374.
%F From _Vladeta Jovovic_, Mar 31 2004: (Start)
%F Coefficient of x^(n-1) in expansion of ((1+x)/(1-x))^n, n > 0.
%F a(n) = Sum_{k=1..n} binomial(n, k)*binomial(n+k-2, k-1), n > 0. (End)
%F D-finite with recurrence (n-1)*(n-2)*a(n) = 3*(2*n-3)*(n-2)*a(n-1) - (n-1)*(n-3)*a(n-2) for n > 2. - _Vladeta Jovovic_, Jul 16 2004
%F a(n+1) = Jacobi_P(n, 1, -1, 3); a(n+1) = Sum{k=0..n} C(n+1, k)*C(n-1, n-k)*2^k. - _Paul Barry_, Jan 23 2006
%F a(n) = n*A006318(n-1) - _Abdullahi Umar_, Aug 25 2008
%F a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n+1) = A035607(2*n,n). - _Reinhard Zumkeller_, Jul 20 2013
%F a(n) = n*hypergeometric([1-n, n], [2], -1) for n >= 1. - _Peter Luschny_, Sep 17 2014
%F O.g.f.: -(x^4 + sqrt(x^2 - 6*x + 1)*(x^3 - 5*x^2 + 5*x + 1) - 8*x^3 + 16*x^2 - 6*x + 1)/(x^3 + sqrt(x^2 - 6*x + 1)*(x^2 - 4*x - 1)- 7*x^2 + 7*x - 1). - _Vladimir Kruchinin_, Nov 25 2016
%F 0 = a(n)*(a(n+1) - 18*a(n+2) + 65*a(n+3) - 12*a(n+4)) + a(n+1)*(54*a(n+2) - 408*a(n+3) + 81*a(n+4)) + a(n+2)*(72*a(n+2) + 334*a(n+3) - 90*a(n+4)) + a(n+3)*(-24*a(n+3) + 9*a(n+4)) for all integer n if a(0) = 0 and a(n) = -2*A050151(-n) for n < 0. - _Michael Somos_, Nov 25 2016
%F O.g.f: (2 - x + x*(3 - x)/sqrt(x^2 - 6*x + 1))/2. - _Petros Hadjicostas_, Feb 14 2021
%F a(n) = A002002(n) - A026002(n-1) for n>=2. - _Shel Kaphan_, Mar 24 2023
%e G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 450*x^5 + 2364*x^6 + 12642*x^7 + ...
%t Flatten[{1,RecurrenceTable[{(n-3)*(n-1)*a[n-2]-3*(n-2)*(2*n-3)*a[n-1]+(n-2)*(n-1)*a[n]==0,a[1]==1,a[2]==4},a,{n,20}]}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%t a[ n_] := If[ n == 0, 1, Sum[ Binomial[n, k] Binomial[n + k - 2, k - 1], {k, n}]]; (* _Michael Somos_, Nov 25 2016 *)
%t a[ n_] := If[ n == 0, 1, Hypergeometric2F1[1 - n, n, 2, -1]; (* _Michael Somos_, Nov 25 2016 *)
%o (PARI) a(n)=if(n==0, 1, sum(k=1,n, binomial(n, k)*binomial(n+k-2, k-1)) ); \\ _Joerg Arndt_, May 04 2013
%o (Haskell)
%o a050146 n = if n == 0 then 1 else a035607 (2 * n - 2) (n - 1)
%o -- _Reinhard Zumkeller_, Nov 05 2013, Jul 20 2013
%o (Sage)
%o A050146 = lambda n : n*hypergeometric([1-n, n], [2], -1) if n>0 else 1
%o [round(A050146(n).n(100)) for n in (0..24)] # _Peter Luschny_, Sep 17 2014
%o (Maxima)
%o taylor(-(x^4+sqrt(x^2-6*x+1)*(x^3-5*x^2+5*x+1)-8*x^3+16*x^2-6*x+1)/(x^3+sqrt(x^2-6*x+1)*(x^2-4*x-1)-7*x^2+7*x-1),x,0,10); /* _Vladimir Kruchinin_, Nov 25 2016
%Y Cf. A002003, A006318, A050151, A008288, A002002, A026002.
%Y -1-diagonal of A266213 for n>=1.
%K nonn
%O 0,3
%A _Clark Kimberling_
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