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

%I A117207 #25 Jan 23 2025 23:31:49
%S A117207 1,3,1,10,7,1,35,31,13,1,126,121,81,21,1,462,456,381,181,31,1,1716,
%T A117207 1709,1583,1058,358,43,1,6435,6427,6231,5055,2605,645,57,1,24310,
%U A117207 24301,24013,21661,14605,5785,1081,73,1,92378,92368,91963,87643,70003,38251,11791
%N A117207 Number triangle read by rows: T(n,k) = Sum_{j=0..n-k} C(n+j,j+k)*C(n-j,k).
%C A117207 Row sums are A037965(n+1).
%C A117207 Second column is A048775. - _Paul Barry_, Oct 01 2010
%C A117207 First column is A001700. - _Dan Uznanski_, Jan 23 2012
%C A117207 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 A117207 Second diagonal is A002061. - _Franklin T. Adams-Watters_, Jan 24 2012
%H A117207 Harvey P. Dale, <a href="/A117207/b117207.txt">Table of n, a(n) for n = 0..1000</a>
%F A117207 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!).
%F A117207 T(n,k) = [x^(n-k)](1+x)^(n-k)*F(-n-1,-n,1,x/(1+x)). - _Paul Barry_, Oct 01 2010
%F A117207 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
%e A117207 Triangle begins:
%e A117207      1,
%e A117207      3,    1,
%e A117207     10,    7,    1,
%e A117207     35,   31,   13,    1,
%e A117207    126,  121,   81,   21,   1,
%e A117207    462,  456,  381,  181,  31,  1,
%e A117207   1716, 1709, 1583, 1058, 358, 43, 1
%t A117207 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 A117207 (PARI) T(n,k)=sum(j=0,n-k, binomial(n+j,j+k)*binomial(n-j,k))
%o A117207 T(n,k)=binomial(2*n+1,n+1)-(n+1)*sum(j=1,k, binomial(n,j-1)^2/j)
%o A117207 A117207(k)=my(n=sqrtint(2*k-sqrtint(2*k))); T(n,k-n*(n+1)/2) \\ _M. F. Hasler_, Jan 25 2012
%K A117207 easy,nonn,tabl
%O A117207 0,2
%A A117207 _Paul Barry_, Mar 02 2006