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%I #35 Feb 16 2017 03:03:57
%S 1,1,1,2,6,1,5,30,15,1,14,140,140,28,1,42,630,1050,420,45,1,132,2772,
%T 6930,4620,990,66,1,429,12012,42042,42042,15015,2002,91,1,1430,51480,
%U 240240,336336,180180,40040,3640,120,1,4862,218790,1312740,2450448,1837836,612612,92820,6120,153,1
%N Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.
%H Indranil Ghosh, <a href="/A280580/b280580.txt">Rows 0..100 of triangle, flattened</a>
%F T(n,k) = A001263(n+1,k+1)*A000108(n)/A000108(k) for 0 <= k <= n.
%F T(n,k) = binomial(2*n,2*k)*A000108(n-k) for 0 <= k <= n.
%F T(n,k) = A039599(n,k)*binomial(n+1+k,2*k+1)/(n+1-k) for 0 <= k <= n.
%F Recurrences: T(n,0) = A000108(n) and (1) T(n,k) = T(n,k-1)*(n+1-k)*(n+2-k)/ (2*k*(2*k-1)) for 0 < k <= n, (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) satisfy the recurrence equation p"(n,x) = 2*n*(2*n-1)*p(n-1,x) with initial value p(0,x) = 1 ( n > 0, p" is the second derivative of p ), and Sum_{n>=0} p(n,x)*t^(2*n)/((2*n)!) = cosh(x*t)*(Sum_{n>=0} A000108(n)*t^(2*n)/((2*n)!)).
%F Conjectures: (1) Sum_{k=0..n} (-1)^k*T(n,k)*A238390(k) = A000007(n);
%F (2) Antidiagonal sums equal A001003(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) = A082758(n).
%F Sum_{k=0..n} T(n,k)*A000108(k) = A000108(n)*A000108(n+1) = A005568(n).
%F Matrix product: Sum_{i=0..n} T(n,i)*T(i,k) = T(n,k)*A000108(n+1-k) for 0<=k<=n.
%F T(n,k) = A097610(2*n,2*k) for 0 <= k <= n.
%F Sum_{k=0..n} (k+1)*T(n,k)*A000108(k) = binomial(2*n+1,n)*A000108(n).
%e Triangle begins:
%e n\k: 0 1 2 3 4 5 6 7 8 . . .
%e 0: 1
%e 1: 1 1
%e 2: 2 6 1
%e 3: 5 30 15 1
%e 4: 14 140 140 28 1
%e 5: 42 630 1050 420 45 1
%e 6: 132 2772 6930 4620 990 66 1
%e 7: 429 12012 42042 42042 15015 2002 91 1
%e 8: 1430 51480 240240 336336 180180 40040 3640 120 1
%e etc.
%e T(3,2) = binomial(6,4) * binomial(2,1) / (3+1-2) = 15 * 2 / 2 = 15. - _Indranil Ghosh_, Feb 15 2017
%Y Row sums are A026945.
%Y Triangle related to A000108, A001006, A001263, and A039599.
%Y Cf. A000007, A001003, A005568, A103365, A238390.
%K nonn,tabl
%O 0,4
%A _Werner Schulte_, Jan 05 2017