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%I #19 Oct 01 2017 03:05:56
%S 0,1,14,75,265,735,1736,3654,7050,12705,21670,35321,55419,84175,
%T 124320,179180,252756,349809,475950,637735,842765,1099791,1418824,
%U 1811250,2289950,2869425,3565926,4397589,5384575,6549215,7916160,9512536
%N Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).
%C All terms are positive: A093566 >= A054563 ==> C( C(n,2), 3) >= C( C(n,3), 2) ==> n^2*(n^4 + 3n^3 -35n^2 + 69n -38)/144 >= 0 ==> (n - 2)(n - 1)(n^2 + 6n - 19) ==> 0 which it is for all n >= 2.
%H Solomon W. Golomb, <a href="http://www.jstor.org/stable/2321859">Iterated binomial coefficients</a>, Amer. Math. Monthly, 87 (1980), 719-727.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = A093566(n) - A054563(n).
%F G.f.: x^3*(-1-7*x+2*x^2+x^3)/(x-1)^7. - _R. J. Mathar_, Dec 08 2010
%p A093567:=n->binomial(binomial(n, 2), 3) - binomial(binomial(n, 3), 2); seq(A093567(n), n=2..30); # _Wesley Ivan Hurt_, Feb 02 2014
%t Table[ Binomial[ Binomial[n, 2], 3] - Binomial[ Binomial[n, 3], 2], {n, 2, 34}]
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,14,75,265,735,1736},40] (* _Harvey P. Dale_, Jun 12 2016 *)
%o (PARI) a(n) = binomial(binomial(n,2), 3) - binomial(binomial(n,3), 2); \\ _Michel Marcus_, Oct 01 2017
%Y Cf. A054563, A093566.
%K nonn
%O 2,3
%A _Robert G. Wilson v_ and _Santi Spadaro_, Mar 31 2004
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