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%I #22 Apr 23 2016 08:18:05
%S 1,3,1,10,7,1,35,31,13,1,126,121,81,21,1,462,456,381,181,31,1,1716,
%T 1709,1583,1058,358,43,1,6435,6427,6231,5055,2605,645,57,1,24310,
%U 24301,24013,21661,14605,5785,1081,73,1,92378,92368,91963,87643,70003,38251,11791
%N Number triangle read by rows: T(n,k)=sum{j=0..n-k, C(n+j,j+k)C(n-j,k)}.
%C Row sums are A037965(n+1).
%C Second column is A048775. - _Paul Barry_, Oct 01 2010
%C First column is A001700. - _Dan Uznanski_, Jan 23 2012
%C 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
%C Second diagonal is A002061. - _Franklin T. Adams-Watters_, Jan 24 2012
%H Harvey P. Dale, <a href="/A117207/b117207.txt">Table of n, a(n) for n = 0..1000</a>
%F 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).
%F T(n,k)=[x^(n-k)](1+x)^(n-k)*F(-n-1,-n,1,x/(1+x)). - _Paul Barry_, Oct 01 2010
%F 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
%e Triangle begins
%e 1,
%e 3, 1,
%e 10, 7, 1,
%e 35, 31, 13, 1,
%e 126, 121, 81, 21, 1,
%e 462, 456, 381, 181, 31, 1,
%e 1716, 1709, 1583, 1058, 358, 43, 1
%t 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 *)
%o (PARI) T(n,k)=sum(j=0,n-k, binomial(n+j,j+k)*binomial(n-j,k))
%o T(n,k)=binomial(2*n+1,n+1)-(n+1)*sum(j=1,k, binomial(n,j-1)^2/j)
%o A117207(k)=my(n=sqrtint(2*k-sqrtint(2*k))); T(n,k-n*(n+1)/2) \\ _M. F. Hasler_, Jan 25 2012
%K easy,nonn,tabl
%O 0,2
%A _Paul Barry_, Mar 02 2006
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