A110608 - OEIS

%I #14 Sep 25 2018 15:01:49

%S 1,2,1,6,8,1,20,45,18,1,70,224,168,32,1,252,1050,1200,450,50,1,924,

%T 4752,7425,4400,990,72,1,3432,21021,42042,35035,12740,1911,98,1,12870,

%U 91520,224224,244608,127400,31360,3360,128,1,48620,393822,1145664,1559376

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

%C First column is A000984. Second column is A110609 = n^2*A000108. Row sums are A005809.

%H G. C. Greubel, <a href="/A110608/b110608.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F From _Peter Bala_, Oct 13 2015: (Start)

%F n-th row polynomial R(n,t) = [x^n] ( (1 + t*x)*(1 + x)^2 )^n.

%F Cf. A008459, whose n-th row polynomial is equal to [x^n] ( (1 + t*x)*(1 + x) )^n.

%F 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 Triangle begin

%e n\k|   0     1     2    3   4   5

%e ---------------------------------

%e 0  |   1

%e 1  |   2     1

%e 2  |   6     8     1

%e 3  |  20    45    18    1

%e 4  |  70   224   168   32   1

%e 5  | 252  1050  1200  450  50   1

%e ...

%t Flatten[Table[Table[Binomial[n,k]Binomial[2n,n-k],{k,0,n}],{n,0,10}]] (* _Harvey P. Dale_, Aug 10 2011 *)

%o (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 (Maxima)

%o 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 C(x,y):=-B(x,y)-(x*(3*y-3)+1)/(9*B(x,y))-1/3;

%o A(x,y):=x*diff(C(x,y),x)*(-1/C(x,y)+1/(1+C(x,y)));

%o taylor(A(x,y),x,0,7,y,0,7); /* _Vladimir Kruchinin_, Sep 24 2018 */

%Y Cf. A000108, A000984, A005809 (row sums), A008459, A110609 (column 2), A120986.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Jul 30 2005