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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"708884a1bbdf120f7fbc384b27eae7b4f5af026af6b2c9ffc2d6b2804453b66e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-65034f179833a7f6","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 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x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 30 + 2*x; v[2] = 700 + 120*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(200 + 20*x + x^2)*v[i - 2] - (100*x + 10*x^2)*deriv(v[i - 2], x) + (30 + 2*x)*v[i - 1] + 10*x*deriv(v[i - 1], x)); 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-65034f179833a7f6","steps":["D"],"edges":["e-aec3072bbd4c8889"],"text":"A075505 \u2192 D"},"headline":"A075505 \u2192 D","chain":null}