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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":"d63a5263b92df3438b84cdfea375e1e436196459d44ff73b2ac6a493873110e8","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-2c8ebdd6fbbf3af3","role":"sequence","name":"shift_x_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.","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:       1;\n   3:       6,        2;\n   4:      34,       24,        3;\n   5:     208,      224,       60,       4;\n   6:    1396,     2000,      840,     120,       5;\n   7:   10232,    18152,    10440,    2360,     210,      6;\n   8:   81320,   171136,   124908,   39200,    5530,    336,     7;\n   9:  696192,  1689088,  1491168,  602784,  118720,  11424,   504,   8;\n  10: 6382864, 17491584, 18079680, 8965824, 2292360, 308448, 21504, 720, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6+4*x+x^2)*P(n-2,x) - (4+6*x+2*x^2)*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(6 + 4*x + x^2)*P[n - 2, x] - (4 + 6*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 2*x)*P[n - 1, x] + (2 + 2*x)*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] = 1; for(n = 3, N, my(i = n); v[i] = -(6 + 4*x + x^2)*v[i - 2] - (4 + 6*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 2*x)*v[i - 1] + (2 + 2*x)*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-d55b2cfe05a04774","steps":["x \u21a6 x+1","D"],"edges":["e-25255a5163ba8ab9","e-a62b22138cce208e"],"text":"A039755 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A039755 \u2192 x \u21a6 x+1 \u2192 D","chain":null}