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

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

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)

Example

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
...

Mathematica

Flatten[Table[Table[Binomial[n, k]Binomial[2n, n-k], {k, 0, n}], {n, 0, 10}]] (* Harvey P. Dale, Aug 10 2011 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n, k)*binomial(2*n, n-k), ", "))) \\ G. C. Greubel, Sep 01 2017
(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 */

Crossrefs

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

Sequence in context: A193734 A318390 A319511 * A318397 A190015 A112007

Adjacent sequences: A110605 A110606 A110607 * A110609 A110610 A110611

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 30 2005

Status

approved