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A105868 Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).

%I A105868 #40 Sep 08 2022 08:45:17
%S A105868 1,0,1,0,2,1,0,0,6,1,0,0,6,12,1,0,0,0,30,20,1,0,0,0,20,90,30,1,0,0,0,
%T A105868 0,140,210,42,1,0,0,0,0,70,560,420,56,1,0,0,0,0,0,630,1680,756,72,1,0,
%U A105868 0,0,0,0,252,3150,4200,1260,90,1,0,0,0,0,0,0,2772,11550,9240,1980,110,1,0,0
%N A105868 Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).
%C A105868 Row sums are the central trinomial coefficients A002426.
%C A105868 Product of A007318 and this sequence is A008459.
%C A105868 Coefficient array for polynomials P(n,x) = x^n*F(1/2-n/2,-n/2;1;4/x). - _Paul Barry_, Oct 04 2008
%C A105868 Column sums give A001850. It appears that the sums along the antidiagonals of the triangle produce A182883. - _Peter Bala_, Mar 06 2013
%H A105868 G. C. Greubel, <a href="/A105868/b105868.txt">Table of n, a(n) for the first 50 rows</a>
%F A105868 G.f.: 1/(sqrt((1-x*y)^2-4*x^2*y)). - _Vladimir Kruchinin_, Oct 28 2020
%e A105868 Triangle begins
%e A105868   1;
%e A105868   0,  1;
%e A105868   0,  2,  1;
%e A105868   0,  0,  6,  1;
%e A105868   0,  0,  6, 12,  1;
%e A105868   0,  0,  0, 30, 20, 1;
%p A105868 gf := 1/((1 - x*y)^2 - 4*y^2*x)^(1/2):
%p A105868 yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):
%p A105868 row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..n):
%p A105868 seq(row(n), n=0..7); # _Peter Luschny_, Oct 28 2020
%t A105868 Flatten[Table[Binomial[n,k]Binomial[k,n-k],{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Nov 12 2014 *)
%o A105868 (Magma) [[Binomial(n,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jun 14 2015
%Y A105868 Cf. A063007. A001850 (column sums), A182883.
%K A105868 easy,nonn,tabl
%O A105868 0,5
%A A105868 _Paul Barry_, Apr 23 2005