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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":"558be5522ab6d0bf2a06fde1117b57ac5b02ea86e5233e22e53d5464835b50cb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ca8f77e380f6014f","role":"sequence","name":"basis_map_A013610 of Riordan array (1-x,x/(1-x)).","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:     7,     24,       9;\n  1:    17,     87,     117,      27;\n  2:    40,    270,     603,     486,       81;\n  3:    92,    768,    2367,    3267,     1863,      243;\n  4:   208,   2064,    8064,   15660,    15471,     6804,      729;\n  5:   464,   5328,   25128,   62640,    87885,    67311,    24057,     2187;\n  6:  1024,  13344,   73584,  223344,   405000,   442746,   276291,    83106,     6561;\n  7:  2240,  32640,  205776,  734832,  1630692,  2313360,  2066715,  1086939,   282123,    19683;\n  8:  4864,  78336,  555264, 2277504,  5969376, 10396512, 12083904,  9115416,  4139991,   944784,   59049;\n  9: 10496, 185088, 1456128, 6739200, 20349792, 41967072, 60011280, 58996512, 38480265, 15372423, 3129597, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n+11+(9*n+18)*x+9*x^2)*P(n,x) = (6*n+28+(27*n+96)*x+(27*n+99)*x^2+27*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 7 + 24*x + 9*x^2.","mathematica":"P[0, x_] := 7 + 24*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 28 + (27*n + 96)*x + (27*n + 99)*x^2 + 27*x^3)*P[n - 1, x])/(3*n + 11 + (9*n + 18)*x + 9*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 7 + 24*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 28 + (27*n + 96)*x + (27*n + 99)*x^2 + 27*x^3)*v[i - 1])/(3*n + 11 + (9*n + 18)*x + 9*x^2)); 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-7717563017cb9e5f","steps":["basis A013610","D+1"],"edges":["e-02e979b2d9fb24b7","e-29b5a9b751a75437"],"text":"A159854 \u2192 basis A013610 \u2192 D+1"},"headline":"A159854 \u2192 basis A013610 \u2192 D+1","chain":null}