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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":"3f780c4df3a58b681b5ce785ce529624de79c5f50598854150f92f7a92ad8bd4","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-126effb6ce513816","role":"sequence","name":"basis_map_A013611 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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:       1;\n  1:       5,        4;\n  2:      14,       44,        16;\n  3:      50,      236,       288,        64;\n  4:     224,     1320,      2480,      1664,       256;\n  5:    1216,     8296,     19920,     20800,      8960,      1024;\n  6:    7776,    59024,    167104,    226240,    152320,     46080,      4096;\n  7:   57312,   472496,   1509760,   2440256,   2150400,   1017856,    229376,    16384;\n  8:  478656,  4218304,  14808512,  27293952,  28971264,  18063360,   6365184,  1114112,   65536;\n  9: 4469184, 41632704, 157709760, 322104320, 392904960, 297034752, 138485760, 37847040, 5308416, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 8*x)*v[i - 1]); 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-b9fbfb57d9269e2f","steps":["basis A013611","D+1"],"edges":["e-0a0b77ef6a10db4c","e-887713b2dc5c69e0"],"text":"A132393 \u2192 basis A013611 \u2192 D+1"},"headline":"A132393 \u2192 basis A013611 \u2192 D+1","chain":null}