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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":"0b581a292d11872ed2fc8bdd42e7944ecf2eaf13e604cb9da15847d000665e3f","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-f8c44f01f6cd4a34","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+3x).","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:       0;\n  1:       6;\n  2:      42,       72;\n  3:     234,      792,       648;\n  4:    1170,     5904,      9720,      5184;\n  5:    5508,    36936,     91368,     98496,      38880;\n  6:   24948,   208656,    688824,   1119744,     894240,     279936;\n  7:  109998,  1102248,   4550904,   9896256,   11936160,    7558272,    1959552;\n  8:  475470,  5552064,  27517320,  74965824,  121111200,  115893504,   60746112,  13436928;\n  9: 2024190, 26988552, 156093480, 511132032, 1035607680, 1328296320, 1052279424, 470292480, 90699264;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-8*n+6-(12*n-12)*x)*P(n,x) = -(4*n^2-46*n+90-(60*n-228)*x+144*x^2)*P(n-1,x) + (7*n-28+(20*n-122)*x+(12*n-168)*x^2-72*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 46*n + 90 - (60*n - 228)*x + 144*x^2)*P[n - 1, x] + (7*n - 28 + (20*n - 122)*x + (12*n - 168)*x^2 - 72*x^3)*D[P[n - 1, x], x])/(2*n^2 - 8*n + 6 - (12*n - 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 - 46*n + 90 - (60*n - 228)*x + 144*x^2)*v[i - 1] + (7*n - 28 + (20*n - 122)*x + (12*n - 168)*x^2 - 72*x^3)*deriv(v[i - 1], x))/(2*n^2 - 8*n + 6 - (12*n - 12)*x)); 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-016d7724d51f6416","steps":["basis A013609","D"],"edges":["e-2339e629d7f9e0b9","e-2b33fd5fca405a6d"],"text":"A099093 \u2192 basis A013609 \u2192 D"},"headline":"A099093 \u2192 basis A013609 \u2192 D","chain":null}