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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":"daa3f7f2873fb2b3114c47aa7703877bda35a5897208315b6b17910c72dce1c7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f5630447d9b46bac","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n.","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:        1;\n  1:        8,         3;\n  2:       64,        57,         9;\n  3:      458,       711,       270,        27;\n  4:     3016,      7089,      4698,      1107,        81;\n  5:    18782,     61107,     61200,     24813,      4212,       243;\n  6:   112564,    476397,    661230,    401139,    115668,     15309,      729;\n  7:   656378,   3455535,   6266484,   5237109,   2237382,    497907,    53946,    2187;\n  8:  3750256,  23757177,  53933994,  58730859,  34323588,  11222469,  2028078,  185895,   6561;\n  9: 21094262, 156778443, 431423496, 587909421, 445630410, 198030663, 52216812, 7932249, 629856, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(100-60*x+9*x^2)*P(n-4,x) + (140+18*x-18*x^2)*P(n-3,x) - (69+36*x+9*x^2)*P(n-2,x) + (14+6*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8 + 3*x, P(2,x) = 64 + 57*x + 9*x^2, P(3,x) = 458 + 711*x + 270*x^2 + 27*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 3*x;\nP[2, x_] := 64 + 57*x + 9*x^2;\nP[3, x_] := 458 + 711*x + 270*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[-(100 - 60*x + 9*x^2)*P[n - 4, x] + (140 + 18*x - 18*x^2)*P[n - 3, x] - (69 + 36*x + 9*x^2)*P[n - 2, x] + (14 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 3*x; v[3] = 64 + 57*x + 9*x^2; v[4] = 458 + 711*x + 270*x^2 + 27*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(100 - 60*x + 9*x^2)*v[i - 4] + (140 + 18*x - 18*x^2)*v[i - 3] - (69 + 36*x + 9*x^2)*v[i - 2] + (14 + 6*x)*v[i - 1]); 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-5733d79f45a14084","steps":["basis A008288","D+1"],"edges":["e-28556ceb40718bbf","e-8793e585d7bcda6c"],"text":"A013620 \u2192 basis A008288 \u2192 D+1"},"headline":"A013620 \u2192 basis A008288 \u2192 D+1","chain":null}