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A091144 a(n) = binomial(n^2, n)/(1+(n-1)*n).

%I A091144 #27 Sep 08 2022 08:45:12
%S A091144 1,1,2,12,140,2530,62832,1997688,77652024,3573805950,190223180840,
%T A091144 11502251937176,779092434772236,58448142042957576,4811642166029230560,
%U A091144 431306008583779517040,41820546066482630185200
%N A091144 a(n) = binomial(n^2, n)/(1+(n-1)*n).
%C A091144 Diagonal of array T(n,k) = binomial(kn,n)/(1+(k-1)n).
%C A091144 Number of paths up and left from (0,0) to (n^2-n,n) where x/y <= n-1 for all intermediate points. - _Henry Bottomley_, Dec 25 2003
%C A091144 Empirical: In the ring of symmetric functions over the fraction field Q(q, t), letting s(1^n) denote the Schur function indexed by (1^n), a(n) is equal to the coefficient of s(n) in nabla^(n)s(1^n) with q=t=1, where nabla denotes the "nabla operator" on symmetric functions, and s(n) denotes the Schur function indexed by the integer partition (n) of n. - _John M. Campbell_, Apr 06 2018
%H A091144 Vincenzo Librandi, <a href="/A091144/b091144.txt">Table of n, a(n) for n = 0..200</a>
%H A091144 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344.
%F A091144 From _Henry Bottomley_, Dec 25 2003: (Start)
%F A091144 a(n) = A014062(n)/A002061(n);
%F A091144 a(n) = A062993(n-2, n);
%F A091144 a(n) = A070914(n, n-1);
%F A091144 a(n) = A071201(n, n^2-n);
%F A091144 a(n) = A071201(n, n^2-n+1);
%F A091144 a(n) = A071202(n, n^2-n+1). (End)
%p A091144 A091144 := proc(n)
%p A091144     binomial(n^2,n)/(1+n*(n-1)) ;
%p A091144 end proc: # _R. J. Mathar_, Feb 14 2015
%t A091144 Table[Binomial[n^2, n] / (n (n - 1) + 1), {n, 0, 20}] (* _Vincenzo Librandi_, Apr 07 2018 *)
%o A091144 (PARI) a(n) = binomial(n^2, n)/(n*(n-1)+1); \\ _Altug Alkan_, Apr 06 2018
%o A091144 (Magma) [Binomial(n^2, n)/(1+(n-1)*n): n in [0..20]]; // _Vincenzo Librandi_, Apr 07 2018
%o A091144 (GAP) List([0..20],n->Binomial(n^2,n)/(1+(n-1)*n)); # _Muniru A Asiru_, Apr 08 2018
%Y A091144 Cf. A002061, A014062, A062993, A070914, A071201, A071202.
%K A091144 nonn,easy
%O A091144 0,3
%A A091144 _Paul Barry_, Dec 22 2003