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A380128 Triangle read by rows: Riordan array (1/(C(x)*sqrt(1-4*x)), x/C(x)) where C(x) is g.f. of A000108.

%I A380128 #6 Jan 25 2025 12:57:24
%S A380128 1,1,1,3,0,1,10,1,-1,1,35,4,0,-2,1,126,15,1,0,-3,1,462,56,5,0,1,-4,1,
%T A380128 1716,210,21,1,0,3,-5,1,6435,792,84,6,0,0,6,-6,1,24310,3003,330,28,1,
%U A380128 0,-1,10,-7,1,92378,11440,1287,120,7,0,0,-4,15,-8,1,352716,43758,5005,495,36,1,0,0,-10,21,-9,1
%N A380128 Triangle read by rows: Riordan array (1/(C(x)*sqrt(1-4*x)), x/C(x)) where C(x) is g.f. of A000108.
%F A380128 G.f.: 1/((C(t) - x*t) * sqrt(1 - 4*t)) where C(t) is g.f. of A000108.
%F A380128 Inverse Riordan array is (2 - D(x), x * D(x)) where D(x) is g.f. of A001764.
%F A380128 Conjecture: T(n, k) = Sum_{i=0..n-k} (-1)^i * binomial(2*n - 2*k, n - k - i) * binomial(k + i, i).
%e A380128 Triangle T(n, k) for 0 <= k <= n starts:
%e A380128 n \k :       0      1     2    3   4   5   6   7    8   9  10  11
%e A380128 =================================================================
%e A380128    0 :       1
%e A380128    1 :       1      1
%e A380128    2 :       3      0     1
%e A380128    3 :      10      1    -1    1
%e A380128    4 :      35      4     0   -2   1
%e A380128    5 :     126     15     1    0  -3   1
%e A380128    6 :     462     56     5    0   1  -4   1
%e A380128    7 :    1716    210    21    1   0   3  -5   1
%e A380128    8 :    6435    792    84    6   0   0   6  -6    1
%e A380128    9 :   24310   3003   330   28   1   0  -1  10   -7   1
%e A380128   10 :   92378  11440  1287  120   7   0   0  -4   15  -8   1
%e A380128   11 :  352716  43758  5005  495  36   1   0   0  -10  21  -9   1
%e A380128   etc.
%o A380128 (PARI) T(n, k) = polcoef(polcoef(1/((C(t)-x*t)*sqrt(1-4*t))+x*O(x^k), k, x)+t*O(t^n), n, t);
%o A380128        m = matrix(15, 15, n, k, if(k>n, 0, T(n-1, k-1)))
%Y A380128 Cf. A088218 (column 0), A001791 (column 1), A000007 (central terms).
%Y A380128 Cf. A000108, A001764.
%K A380128 sign,easy,tabl
%O A380128 0,4
%A A380128 _Werner Schulte_, Jan 12 2025