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an exact degree step keeps every row nonzero; S_n = (A + B D + C D^2 with A = 12 + 14 n + 4 n^2 + 45 x + 57 n x + 18 n^2 x, B = 13 x + 8 n x + 15 x^2 + 9 n x^2, C = 0) / (4 n^2 + 14 n + 12): Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. 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Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 10*x + 15*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 14*n + 12 + (18*n^2 + 57*n + 45)*x)*v[i - 1] + ((8*n + 13)*x + (9*n + 15)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-24790f2ac2b21c79","steps":[],"edges":[],"text":"A370258"},"headline":"Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= n.","chain":null}