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%I #17 Nov 25 2018 18:08:55
%S 1,3,1,10,10,1,35,70,21,1,126,420,252,36,1,462,2310,2310,660,55,1,
%T 1716,12012,18018,8580,1430,78,1,6435,60060,126126,90090,25025,2730,
%U 105,1,24310,291720,816816,816816,340340,61880,4760,136,1,92378,1385670,4988412,6651216,3879876,1058148,135660,7752,171,1
%N Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.
%H Indranil Ghosh, <a href="/A281000/b281000.txt">Rows 0..100 of triangle, flattened</a>
%F T(n,k) = A097610(2*n+1, 2*k+1) = binomial(2*n+1, 2*k+1)*A000108(n-k) = A280580(n,k)*(2*n+1)/(2*k+1) for 0 <= k <= n.
%F Recurrences: T(n,0) = (2*n+1)*A000108(n) and
%F (1) T(n,k) = T(n,k-1)*(n+1-k)*(n+2-k)/(2*k*(2*k+1)) for 0 < k <= n,
%F (2) T(n,k) = T(n-1, k-1)*n*(2*n+1)/(k*(2*k+1)).
%F The row polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^(2*k+1) satisfy the recurrence equation p"(n,x) = (2*n+1)*2*n*p(n-1,x) with initial value p(0,x) = x (n > 0, p" is the second derivative of p), and Sum_{n>=0} p(n,x)*t^(2*n+1)/ ((2*n+1)!) = sinh(x*t)*(Sum_{n>=0} A000108(n)*t^(2*n)/((2*n)!)).
%F Conjectures:
%F (1) Antidiagonal sums equal A001003(n+1);
%F (2) Sum_{k=0..n} (-1)^k*T(n,k)*(2*k+1)*A000108(k)*A103365(k) = A000007(n);
%F (3) Matrix inverse equals T(n,k)*A103365(n+1-k).
%F Sum_{k=0..n} (n+1-k)*T(n,k) = A002426(2*n+1) = A273055(n).
%F Sum_{k=0..n} T(n,k)*(2*k+1)*A000108(k) = (2*n+1)*A000108(n)*A000108(n+1) = A125558(n+1).
%F Matrix product: Sum_{i=0..n} T(n,i)*T(i,k) = T(n,k)*A000108(n+1-k) for 0<=k<=n.
%e Triangle begins:
%e n\k: 0 1 2 3 4 5 6 7 8 . . .
%e 0: 1
%e 1: 3 1
%e 2: 10 10 1
%e 3: 35 70 21 1
%e 4: 126 420 252 36 1
%e 5: 462 2310 2310 660 55 1
%e 6: 1716 12012 18018 8580 1430 78 1
%e 7: 6435 60060 126126 90090 25025 2730 105 1
%e 8: 24310 291720 816816 816816 340340 61880 4760 136 1
%e etc.
%e T(3,2) = binomial(7,5) * binomial(2,1) / (3+1-2) = 21 * 2 / 2 = 21. - _Indranil Ghosh_, Feb 15 2017
%t Table[Binomial[2n+1,2k+1] Binomial[2n-2k,n-k]/(n+1-k),{n,0,10},{k,0,n}]// Flatten (* _Harvey P. Dale_, Nov 25 2018 *)
%Y Row sums are A099250.
%Y Triangle related to A000108, A097610, A280580.
%Y Cf. A000007, A001003, A001006, A002426, A103365, A125558, A273055.
%K nonn,easy,tabl
%O 0,2
%A _Werner Schulte_, Jan 12 2017
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