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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":"a2cdc7006eddb1c4265b69bdf439d2f036b82ffddc8ce968cce97fe922bdab71","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-7c95902cf1293962","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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:           6,           4;\n  2:          38,          60,          16;\n  3:         344,         744,         432,          64;\n  4:        4104,       10616,        8688,        2688,         256;\n  5:       60976,      177936,      178880,       79040,       15360,       1024;\n  6:     1085872,     3463440,     4011264,     2211520,      618240,      82944,       4096;\n  7:    22555200,    77118016,    99253056,    63899136,    22516480,    4365312,     430080,     16384;\n  8:   535524800,  1937400768,  2708795584,  1959862016,   814082304,  200359936,   28614656,   2162688,    65536;\n  9: 14309014400, 54271248768, 81174724992, 64404509952, 30312221184, 8855626752, 1614004224, 177242112, 10616832, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(24*n-32)*x+16*x^2)*P(n-2,x) + (6*n-5+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (24*n - 32)*x + 16*x^2)*P[n - 2, x] + (6*n - 5 + 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] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (24*n - 32)*x + 16*x^2)*v[i - 2] + (6*n - 5 + 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-260e022530e5976c","steps":["basis A013611","D+1"],"edges":["e-10584eb473514fc3","e-a2b3da09b8593cd1"],"text":"A286718 \u2192 basis A013611 \u2192 D+1"},"headline":"A286718 \u2192 basis A013611 \u2192 D+1","chain":null}