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A110608 Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).

%I A110608 #14 Sep 25 2018 15:01:49
%S A110608 1,2,1,6,8,1,20,45,18,1,70,224,168,32,1,252,1050,1200,450,50,1,924,
%T A110608 4752,7425,4400,990,72,1,3432,21021,42042,35035,12740,1911,98,1,12870,
%U A110608 91520,224224,244608,127400,31360,3360,128,1,48620,393822,1145664,1559376
%N A110608 Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).
%C A110608 First column is A000984. Second column is A110609 = n^2*A000108. Row sums are A005809.
%H A110608 G. C. Greubel, <a href="/A110608/b110608.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A110608 From _Peter Bala_, Oct 13 2015: (Start)
%F A110608 n-th row polynomial R(n,t) = [x^n] ( (1 + t*x)*(1 + x)^2 )^n.
%F A110608 Cf. A008459, whose n-th row polynomial is equal to [x^n] ( (1 + t*x)*(1 + x) )^n.
%F A110608 exp( Sum_{n >= 1} R(n,t)*x^n/n ) = 1 + (2 + t)*x + (5 + 6*t + t^2)*x^2 + ... is the o.g.f. for A120986. (End)
%e A110608 Triangle begin
%e A110608 n\k|   0     1     2    3   4   5
%e A110608 ---------------------------------
%e A110608 0  |   1
%e A110608 1  |   2     1
%e A110608 2  |   6     8     1
%e A110608 3  |  20    45    18    1
%e A110608 4  |  70   224   168   32   1
%e A110608 5  | 252  1050  1200  450  50   1
%e A110608 ...
%t A110608 Flatten[Table[Table[Binomial[n,k]Binomial[2n,n-k],{k,0,n}],{n,0,10}]] (* _Harvey P. Dale_, Aug 10 2011 *)
%o A110608 (PARI) for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(2*n,n-k), ", "))) \\ _G. C. Greubel_, Sep 01 2017
%o A110608 (Maxima)
%o A110608 B(x,y):=(sqrt(-x*(4*x^2*y^3+(-12*x^2-8*x)*y^2+(12*x^2-20*x+4)*y-4*x^2+x))/(2*3^(3/2))-(x*(18*y+9)-2)/54)^(1/3);
%o A110608 C(x,y):=-B(x,y)-(x*(3*y-3)+1)/(9*B(x,y))-1/3;
%o A110608 A(x,y):=x*diff(C(x,y),x)*(-1/C(x,y)+1/(1+C(x,y)));
%o A110608 taylor(A(x,y),x,0,7,y,0,7); /* _Vladimir Kruchinin_, Sep 24 2018 */
%Y A110608 Cf. A000108, A000984, A005809 (row sums), A008459, A110609 (column 2), A120986.
%K A110608 easy,nonn,tabl
%O A110608 0,2
%A A110608 _Paul Barry_, Jul 30 2005