A117207 Number triangle read by rows: T(n,k) = Sum_{j=0..n-k} C(n+j,j+k)*C(n-j,k).

1, 3, 1, 10, 7, 1, 35, 31, 13, 1, 126, 121, 81, 21, 1, 462, 456, 381, 181, 31, 1, 1716, 1709, 1583, 1058, 358, 43, 1, 6435, 6427, 6231, 5055, 2605, 645, 57, 1, 24310, 24301, 24013, 21661, 14605, 5785, 1081, 73, 1, 92378, 92368, 91963, 87643, 70003, 38251, 11791

Offset: 0

Author

Paul Barry, Mar 02 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A037965(n+1).

Second column is A048775. - Paul Barry, Oct 01 2010

First column is A001700. - Dan Uznanski, Jan 23 2012

The number of different ordered partitions of n+1 into n+1 bins (as with A001700), such that more than k bins are nonempty. - Dan Uznanski, Jan 23 2012

Second diagonal is A002061. - Franklin T. Adams-Watters, Jan 24 2012

Examples

			Triangle begins:
     1,
     3,    1,
    10,    7,    1,
    35,   31,   13,    1,
   126,  121,   81,   21,   1,
   462,  456,  381,  181,  31,  1,
  1716, 1709, 1583, 1058, 358, 43, 1
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Programs

Mathematica

Table[Sum[Binomial[n+j,j+k]Binomial[n-j,k],{j,0,n-k}],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Apr 23 2016 *)

PARI

T(n,k)=sum(j=0,n-k, binomial(n+j,j+k)*binomial(n-j,k))
T(n,k)=binomial(2*n+1,n+1)-(n+1)*sum(j=1,k, binomial(n,j-1)^2/j)
A117207(k)=my(n=sqrtint(2*k-sqrtint(2*k))); T(n,k-n*(n+1)/2) \\ M. F. Hasler, Jan 25 2012

Formula

T(n,k) = C(2*n+1,n+1) - (n+1)*Sum_{j=1..k} (Product_{i=0..j-2} (n-i)^2)/((j-1)!*j!).

T(n,k) = [x^(n-k)](1+x)^(n-k)*F(-n-1,-n,1,x/(1+x)). - Paul Barry, Oct 01 2010

T(n,k) = C(2*n+1,n+1) - (n+1)*Sum_{j=1..k} C(n,j-1)^2/j. - M. F. Hasler, Jan 25 2012