OFFSET
0,2
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
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
FORMULA
T(n,k)=C(2n+1,n+1)-sum{j=1..k, product{i=0..j-2, (n-i)^2}/((j-1)!j!)}}*(n+1).
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(2n+1,n+1)-(n+1)*sum(j=1,k, C(n,j-1)^2/j). - M. F. Hasler, Jan 25 2012
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 02 2006
STATUS
approved