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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":"a5cfcae7eaec6ed023824213c99df15ccac9bab3157ccf6865a825efee23c1e8","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-dba956a9ceeed98f","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:            416,             210,              25;\n  1:           7936,            6240,            1575,             125;\n  2:         147456,          158720,           62400,           10500,             625;\n  3:        2686976,         3686400,         1984000,          520000,           65625,           3125;\n  4:       48234496,        80609280,        55296000,        19840000,         3900000,         393750,          15625;\n  5:      855638016,     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25*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((80*n + 416 + (25*n + 210)*x + 25*x^2)*P[n - 1, x])/(5*n + 21 + 5*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 416 + 210*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 416 + (25*n + 210)*x + 25*x^2)*v[i - 1])/(5*n + 21 + 5*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-2bc1923905556bf6","steps":["x \u21a6 x+1","D+1"],"edges":["e-2406b2ec50933490","e-5f3ec16f3a8168d7"],"text":"A038319 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038319 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}