A378783 Triangular array T(n,k) read by rows: T(n, k) = c_k(n+1). The sequence c_k(m) has the ordinary generating function C_k(x) which satisfies C_k(x) = 1/(1+C_k(x)*Sum_{t=0..k} x^(t+1)).
-1, 2, 1, -5, -1, -2, 14, 1, 5, 4, -42, -1, -12, -8, -9, 132, 1, 29, 18, 22, 21, -429, -1, -73, -43, -54, -50, -51, 1430, 1, 190, 105, 135, 124, 128, 127, -4862, -1, -505, -262, -345, -315, -326, -322, -323, 16796, 1, 1363, 666, 896, 813, 843, 832, 836, 835
Offset: 0
Examples
Triangle begins: [0] -1 [1] 2, 1 [2] -5, -1, -2 [3] 14, 1, 5, 4 [4] -42, -1, -12, -8, -9 [5] 132, 1, 29, 18, 22, 21 [6] -429, -1, -73, -43, -54, -50, -51 [7] 1430, 1, 190, 105, 135, 124, 128, 127 [8] -4862, -1, -505, -262, -345, -315, -326, -322, -323 .
Programs
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Mathematica
T[n_,k_]:=SeriesCoefficient[(2 / (Sqrt[1+4*Sum[x^(t+1),{t,0,k}]] + 1) - 1)/x,{x,0,n}];Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *) -
PARI
column(n, max_n) = { my(s = 1,x = 'x,cu); for(k = 0, max_n-1, cu = cu+polcoeff(1/s+O(x^(k+1)), k, x); cu = cu-polcoeff(1/s+O(x^(k+1)), k-1-n, x); s = s+cu*x^(k+1)); Vec(1/s+O(x^max_n)) }; T(n, k) = column(k, n+2)[n+2] T(n, k) = polcoeff(2 / (sqrt(1+4*x*sum(t=0, k, x^t)) + 1) + O(x^(n+2)), n+1, x)