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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":"56d99d159c33132fadd8d886065794245698adaaa64aed020a21bac4aa522828","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-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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:        4;\n   3:       40,         64;\n   4:      316,       1344,        1152;\n   5:     2320,      19840,       41472,        24576;\n   6:    16564,     255360,      979200,      1351680,        614400;\n   7:   116920,    3073984,    19284480,     46571520,      47923200,      17694720;\n   8:   821356,   35684544,   344598912,   1296261120,    2266521600,    1857945600,     578027520;\n   9:  5758240,  405671680,  5811305472,  31980404736,   84536524800,  115440353280,   78611742720,   21139292160;\n  10: 40333924, 4554520320, 94471672320, 731602206720, 2743850188800, 5551541452800, 6208015564800, 3614818959360, 856141332480;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(21+48*x+32*x^2)*P(n-2,x) - (12*x+28*x^2+16*x^3)*d/dx P(n-2,x) + (10+16*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[-(21 + 48*x + 32*x^2)*P[n - 2, x] - (12*x + 28*x^2 + 16*x^3)*D[P[n - 2, x], x] + (10 + 16*x)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = -(21 + 48*x + 32*x^2)*v[i - 2] - (12*x + 28*x^2 + 16*x^3)*deriv(v[i - 2], x) + (10 + 16*x)*v[i - 1] + (4*x + 4*x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f425461cbf5769b3","steps":["D"],"edges":["e-f0e7be966f0a182a"],"text":"A225473 \u2192 D"},"headline":"A225473 \u2192 D","chain":null}