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A046224 Distinct numbers seen when writing first numerator and then denominator of central elements of 1/2-Pascal triangle.

%I A046224 #37 Jul 02 2025 16:01:56
%S A046224 1,2,3,11,40,147,546,2046,7722,29315,111826,428298,1646008,6344366,
%T A046224 24515700,94942620,368404110,1431985635,5574725970,21732560850,
%U A046224 84828633120,331488081210,1296712152060,5077282282020,19897457591700,78039200913102,306302623291476
%N A046224 Distinct numbers seen when writing first numerator and then denominator of central elements of 1/2-Pascal triangle.
%H A046224 G. C. Greubel, <a href="/A046224/b046224.txt">Table of n, a(n) for n = 1..200</a>
%F A046224 a(n) = Sum_{k=1..n-2} (2*k+1)*binomial(2*n-k-5,n-3), n>2; a(1)=1, a(2)=2. - _Vladimir Kruchinin_, Sep 27 2011
%F A046224 a(n) = (5*n-9)/(8*n-12)*binomial(2*n-2,n-1), n>2; a(1)=1, a(2)=2. - _Eric Werley_, Sep 16 2015
%F A046224 G.f.: (3/2)*x^2 + (2*x - 3*x^2)/(2*sqrt(1-4*x)). - _G. C. Greubel_, Sep 24 2015
%e A046224 1/1; <-- hence 1;
%e A046224 1/1 1/1;
%e A046224 1/1 1/2 1/1; <-- hence 2
%e A046224 1/1 3/2 3/2 1/1;
%e A046224 1/1 5/2 3/1 5/2 1/1; <-- hence 3
%e A046224 1/1 7/2 11/2 11/2 7/2 1/1;
%e A046224 1/1 9/2 9/1 11/1 9/1 9/2 1/1; <-- hence 11
%e A046224 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1;
%e A046224 ...
%t A046224 Join[{1, 2}, Table[(5 n - 9)/(8 n - 12) Binomial[2 n - 2, n - 1], {n, 3, 40}]] (* _Vincenzo Librandi_, Sep 24 2015 *)
%o A046224 (Magma) [1,2] cat [(5*n-9)/(8*n-12)*Binomial(2*n-2,n-1): n in [3..40]]; // _Vincenzo Librandi_, Sep 24 2015
%o A046224 (PARI) a(n) = if (n<3, n, (5*n-9)/(8*n-12)*binomial(2*n-2,n-1));
%o A046224 vector(40, n, a(n)) \\ _Altug Alkan_, Oct 01 2015
%Y A046224 Cf. A046213.
%K A046224 nonn
%O A046224 1,2
%A A046224 _Mohammad K. Azarian_
%E A046224 More terms from _James Sellers_, Dec 13 1999
%E A046224 a(26)-a(27) from _Vincenzo Librandi_, Sep 24 2015