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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":"3407495af39654c2d19789710982cb6b0f9547c7c359602a27ae24279708711d","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-ab79cd460b94a5bd","role":"sequence","name":"theta_plus_one of Triangle read by rows: 3-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:      1;\n   2:      5,       2;\n   3:     23,      20,        3;\n   4:    101,     146,       48,        4;\n   5:    431,     932,      507,       92,       5;\n   6:   1805,    5522,     4440,     1320,     155,       6;\n   7:   7463,   31220,    34923,    15160,    2890,     240,      7;\n   8:  30581,  171026,   256368,   152684,   42070,    5628,    350,     8;\n   9: 124511,  916292,  1794747,  1410612,  527415,  101136,  10066,   488,   9;\n  10: 504605, 4830482, 12142920, 12267280, 5982585, 1543122, 218652, 16872, 657, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (24+24*x+8*x^2+x^3)*P(n-3,x) + (6*x+5*x^2+x^3)*d/dx P(n-3,x) - (26+17*x+3*x^2)*P(n-2,x) - (5*x+2*x^2)*d/dx P(n-2,x) + (9+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 23 + 20*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 2*x;\nP[3, x_] := 23 + 20*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(24 + 24*x + 8*x^2 + x^3)*P[n - 3, x] + (6*x + 5*x^2 + x^3)*D[P[n - 3, x], x] - (26 + 17*x + 3*x^2)*P[n - 2, x] - (5*x + 2*x^2)*D[P[n - 2, x], x] + (9 + 3*x)*P[n - 1, x] + 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] = 1; v[2] = 5 + 2*x; v[3] = 23 + 20*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (24 + 24*x + 8*x^2 + x^3)*v[i - 3] + (6*x + 5*x^2 + x^3)*deriv(v[i - 3], x) - (26 + 17*x + 3*x^2)*v[i - 2] - (5*x + 2*x^2)*deriv(v[i - 2], x) + (9 + 3*x)*v[i - 1] + 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-a2b8ff03da0c14a3","steps":["\u03b8+1","D+1"],"edges":["e-8b885dd48ba326e7","e-8e73cbf1207fdaaf"],"text":"A143495 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A143495 \u2192 \u03b8+1 \u2192 D+1","chain":null}