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A360282 Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.

%I A360282 #16 Feb 14 2023 03:51:51
%S A360282 1,0,1,0,3,2,0,10,12,3,0,35,60,30,4,0,126,280,210,60,5,0,462,1260,
%T A360282 1260,560,105,6,0,1716,5544,6930,4200,1260,168,7,0,6435,24024,36036,
%U A360282 27720,11550,2520,252,8
%N A360282 Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.
%C A360282 Inspired by _Vladimir Kruchinin_'s A360546.
%F A360282 The sequence is member of a family of sequences defined by T(0, 0, m) = 1 and for m > 0, n > 0 as T(n, k, m) = (1/m)*binomial(m*u, u)*binomial(u + n - 1, k - 1), where u = n - k + 1. Here the case m = 2 is considered.
%F A360282 G.f.: (y*(1 - x*y)) / (2*sqrt(x*(y*(x*y - 2) - 4) + 1)) - y/2 + 1. - _Vladimir Kruchinin_, Feb 14 2023
%e A360282 Triangle T(n, k) starts:
%e A360282 [0] 1;
%e A360282 [1] 0,     1;
%e A360282 [2] 0,     3,      2;
%e A360282 [3] 0,    10,     12,      3;
%e A360282 [4] 0,    35,     60,     30,      4;
%e A360282 [5] 0,   126,    280,    210,     60,     5;
%e A360282 [6] 0,   462,   1260,   1260,    560,   105,     6;
%e A360282 [7] 0,  1716,   5544,   6930,   4200,  1260,   168,    7;
%e A360282 [8] 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
%e A360282 [9] 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;
%p A360282 T := proc(n, k) if n = 0 then 1 else m := n - k + 1; (1/2) * binomial(2*m, m) * binomial(m + n - 1, k - 1) fi end:
%p A360282 seq(seq(T(n, k), k = 0..n), n = 0..8);
%p A360282 # With _Vladimir Kruchinin_'s g.f.:
%p A360282 gf := 1 + (y - x*y^2)/(2*sqrt((x*y - 1)^2 - 4*x)) ;
%p A360282 serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
%p A360282 seq(print(seq(coeff(poly(n), y, k), k = 0..n)), n = 0..9); # _Peter Luschny_, Feb 14 2023
%Y A360282 Cf. A135503, A088218 (and A001700), A182626 (row sums apart from sign).
%Y A360282 Family: A172431 and unsigned A053123 (case 1), this sequence is case 2, A360546 (case 3).
%K A360282 nonn,tabl
%O A360282 0,5
%A A360282 _Peter Luschny_, Feb 11 2023