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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":"362a57b3a5bb061c2e776ef2ae03ce601236f1e095e3030f6ca1614025e7fd3a","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-b6cbf920aaa38713","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:           4,           1;\n  2:          27,          11,          1;\n  3:         261,         135,         21,         1;\n  4:        3294,        1980,        405,        34,        1;\n  5:       51354,       34344,       8370,       945,       50,       1;\n  6:      953532,      691740,     191079,     26145,     1890,      69,      1;\n  7:    20549052,    15908724,    4835943,    759969,    67410,    3402,     91,    1;\n  8:   504199728,   411885000,  135162432,  23623488,  2430729,  151956,   5670,  116,   1;\n  9: 13877932176, 11864754936, 4147383060, 789346620, 91071540, 6656013, 309960, 8910, 144, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 2*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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 2*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-8229dbdd6a4b5cd9","steps":["x \u21a6 x+1","D+1"],"edges":["e-64ce901fd869872f","e-b8d281caa1e6d755"],"text":"A225470 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A225470 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}