cp's OEIS Frontend

A086614 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.

%I A086614 #8 Nov 02 2021 09:42:41
%S A086614 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 A086614 770,504,132,8,84,504,1650,3080,3276,1848,429,9,120,924,3960,10010,
%U A086614 15288,13860,6864,1430,10,165,1584,8580,28028,57330,73920,58344,25740
%N A086614 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 A086614 T(n,k) = binomial(2*k, k-1)*binomial(n+k+1, n-k) / k for k > 0. # _Peter Luschny_, Jan 26 2018
%e A086614 Rows:
%e A086614 {1},
%e A086614 {2, 1},
%e A086614 {3, 4,    2},
%e A086614 {4, 10,  12,    5},
%e A086614 {5, 20,  42,   40,   14},
%e A086614 {6, 35, 112,  180,  140,   42},
%e A086614 {7, 56, 252,  600,  770,  504,  132},
%e A086614 {8, 84, 504, 1650, 3080, 3276, 1848, 429}, ...
%p A086614 T := (n,k) -> `if`(k=0, n+1, binomial(2*k, k-1)*binomial(n+k+1, n-k)/k):
%p A086614 for n from 0 to 8 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Jan 26 2018
%Y A086614 T(n,n) = A000108(n).
%Y A086614 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 A086614 nonn,tabl,easy
%O A086614 0,2
%A A086614 _Paul D. Hanna_, Jul 24 2003