A091144 - OEIS

%I #27 Sep 08 2022 08:45:12

%S 1,1,2,12,140,2530,62832,1997688,77652024,3573805950,190223180840,

%T 11502251937176,779092434772236,58448142042957576,4811642166029230560,

%U 431306008583779517040,41820546066482630185200

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

%C Diagonal of array T(n,k) = binomial(kn,n)/(1+(k-1)n).

%C 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 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 Vincenzo Librandi, <a href="/A091144/b091144.txt">Table of n, a(n) for n = 0..200</a>

%H 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 From _Henry Bottomley_, Dec 25 2003: (Start)

%F a(n) = A014062(n)/A002061(n);

%F a(n) = A062993(n-2, n);

%F a(n) = A070914(n, n-1);

%F a(n) = A071201(n, n^2-n);

%F a(n) = A071201(n, n^2-n+1);

%F a(n) = A071202(n, n^2-n+1). (End)

%p A091144 := proc(n)

%p     binomial(n^2,n)/(1+n*(n-1)) ;

%p end proc: # _R. J. Mathar_, Feb 14 2015

%t Table[Binomial[n^2, n] / (n (n - 1) + 1), {n, 0, 20}] (* _Vincenzo Librandi_, Apr 07 2018 *)

%o (PARI) a(n) = binomial(n^2, n)/(n*(n-1)+1); \\ _Altug Alkan_, Apr 06 2018

%o (Magma) [Binomial(n^2, n)/(1+(n-1)*n): n in [0..20]]; // _Vincenzo Librandi_, Apr 07 2018

%o (GAP) List([0..20],n->Binomial(n^2,n)/(1+(n-1)*n)); # _Muniru A Asiru_, Apr 08 2018

%Y Cf. A002061, A014062, A062993, A070914, A071201, A071202.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Dec 22 2003