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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":"cce7f1a34795c24793097e0ccc6bf965039f9a8705b5e13a4f95b01c3c093603","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-930be05ae3b321d9","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 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- 2, x] - ((48*n - 60)*x + (64*n^2 - 272*n + 336)*x^2 - (64*n^2 - 416*n + 644)*x^3 - (192*n - 496)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (40*n - 78 + (32*n^2 - 128*n + 206)*x + (80*n - 164)*x^2 + 32*x^3)*P[n - 1, x] + ((16*n - 36)*x - (16*n - 68)*x^2 - 32*x^3)*D[P[n - 1, x], x])/(4*n - 9 + 8*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n - 105 + (96*n^2 - 384*n + 498)*x + (64*n^3 - 368*n^2 + 860*n - 913)*x^2 + (192*n^2 - 736*n + 716)*x^3 + (128*n - 160)*x^4)*v[i - 2] - ((48*n - 60)*x + (64*n^2 - 272*n + 336)*x^2 - (64*n^2 - 416*n + 644)*x^3 - (192*n - 496)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (40*n - 78 + (32*n^2 - 128*n + 206)*x + (80*n - 164)*x^2 + 32*x^3)*v[i - 1] + ((16*n - 36)*x - (16*n - 68)*x^2 - 32*x^3)*deriv(v[i - 1], x))/(4*n - 9 + 8*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-930be05ae3b321d9","steps":["D+1"],"edges":["e-7f2d1aa28c2a754d"],"text":"A257621 \u2192 D+1"},"headline":"A257621 \u2192 D+1","chain":null}