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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":"427f199cdb1540a38ce440efc8538ca11b82a140c75b6664c7a4d88b82f8ff0c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1b336857eed4ab2c","role":"sequence","name":"derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    0;\n   2:    0;\n   3:    2;\n   4:   12,     6;\n   5:   42,    48,    12;\n   6:  112,   216,   120,    20;\n   7:  252,   720,   660,   240,    30;\n   8:  504,  1980,  2640,  1560,   420,   42;\n   9:  924,  4752,  8580,  7280,  3150,  672,   56;\n  10: 1584, 10296, 24024, 27300, 16800, 5712, 1008, 72;\n","recurrence_sentence":"The row polynomials satisfy (2*n-6)*P(n,x) = (2*n+4+(n+2)*x)*P(n-1,x) + (4*x+x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (n + 2)*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((2*n + 4 + (n + 2)*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["D","D"],"edges":["e-1dbefba2139b076a","e-826e3e8d10092205"],"text":"A078812 \u2192 D \u2192 D"},"headline":"A078812 \u2192 D \u2192 D","chain":null}