A110608 Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).
1, 2, 1, 6, 8, 1, 20, 45, 18, 1, 70, 224, 168, 32, 1, 252, 1050, 1200, 450, 50, 1, 924, 4752, 7425, 4400, 990, 72, 1, 3432, 21021, 42042, 35035, 12740, 1911, 98, 1, 12870, 91520, 224224, 244608, 127400, 31360, 3360, 128, 1, 48620, 393822, 1145664, 1559376
Offset: 0
Examples
Triangle begin n\k| 0 1 2 3 4 5 --------------------------------- 0 | 1 1 | 2 1 2 | 6 8 1 3 | 20 45 18 1 4 | 70 224 168 32 1 5 | 252 1050 1200 450 50 1 ...
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Mathematica
Flatten[Table[Table[Binomial[n,k]Binomial[2n,n-k],{k,0,n}],{n,0,10}]] (* Harvey P. Dale, Aug 10 2011 *) -
Maxima
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); C(x,y):=-B(x,y)-(x*(3*y-3)+1)/(9*B(x,y))-1/3; A(x,y):=x*diff(C(x,y),x)*(-1/C(x,y)+1/(1+C(x,y))); taylor(A(x,y),x,0,7,y,0,7); /* Vladimir Kruchinin, Sep 24 2018 */
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PARI
for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(2*n,n-k), ", "))) \\ G. C. Greubel, Sep 01 2017
Formula
From Peter Bala, Oct 13 2015: (Start)
n-th row polynomial R(n,t) = [x^n] ( (1 + t*x)*(1 + x)^2 )^n.
Cf. A008459, whose n-th row polynomial is equal to [x^n] ( (1 + t*x)*(1 + x) )^n.
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)
Comments