Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #5 Sep 13 2013 17:16:43
%S 1,2,2,6,8,6,20,36,30,20,70,160,168,112,70,252,700,900,720,420,252,
%T 924,3024,4620,4400,2970,1584,924,3432,12936,22932,25480,20020,12012,
%U 6006,3432,12870,54912,110880,141120,127400,87360,48048,22880,12870,48620
%N Triangle T(n,k) = binomial(2*n,k) *binomial(2*n-2*k,n-k), read by rows; 0<=k<=n.
%C Row sums are s(n) = 1, 4, 20, 106, 580, 3244,,...
%F Conjecture for row sums: 2*(n+1)*(2*n+1)*s(n) +(-81*n^2+19*n-8)*s(n-1) +10*(51*n^2-77*n+30)*s(n-2) -500*(n-1)*(2*n-3)*s(n-3)=0. - _R. J. Mathar_, Sep 13 2013
%e 1;
%e 2, 2;
%e 6, 8, 6;
%e 20, 36, 30, 20;
%e 70, 160, 168, 112, 70;
%e 252, 700, 900, 720, 420, 252;
%e 924, 3024, 4620, 4400, 2970, 1584, 924;
%e 3432, 12936, 22932, 25480, 20020, 12012, 6006, 3432;
%e 12870, 54912, 110880, 141120, 127400, 87360, 48048, 22880, 12870;
%e 48620, 231660, 525096, 753984, 771120, 599760, 371280, 190944, 87516, 48620';
%e 184756, 972400, 2445300, 3912480, 4476780, 3907008, 2713200, 1550400, 755820, 335920, 184756;
%t t[n_, m_] = (Binomial[2*n, m]*Binomial[2*(n - m), (n - m)]); Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]
%Y Cf. A062344.
%K nonn,tabl
%O 0,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 17 2008