%I #8 Nov 02 2021 09:42:41

%S 1,2,1,3,4,2,4,10,12,5,5,20,42,40,14,6,35,112,180,140,42,7,56,252,600,

%T 770,504,132,8,84,504,1650,3080,3276,1848,429,9,120,924,3960,10010,

%U 15288,13860,6864,1430,10,165,1584,8580,28028,57330,73920,58344,25740

%N Triangle read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.

%F T(n,k) = binomial(2*k, k-1)*binomial(n+k+1, n-k) / k for k > 0. # _Peter Luschny_, Jan 26 2018

%e Rows:

%e {1},

%e {2, 1},

%e {3, 4, 2},

%e {4, 10, 12, 5},

%e {5, 20, 42, 40, 14},

%e {6, 35, 112, 180, 140, 42},

%e {7, 56, 252, 600, 770, 504, 132},

%e {8, 84, 504, 1650, 3080, 3276, 1848, 429}, ...

%p T := (n,k) -> `if`(k=0, n+1, binomial(2*k, k-1)*binomial(n+k+1, n-k)/k):

%p for n from 0 to 8 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Jan 26 2018

%Y T(n,n) = A000108(n).

%Y Cf. A086615 (antidiagonal sums), A086616 (row sums), A086617, A000292 (column 1), A277935 (column 2), A000580 (column 3 divided by 5), A000582 (column 4 divided by 14).

%K nonn,tabl,easy

%O 0,2

%A _Paul D. Hanna_, Jul 24 2003