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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":"22ace7beaa3f5393278e51eb1f6a3887616131114b42e7eb321da8ec19aa0baa","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-a33b47c8b957d83a","role":"sequence","name":"basis_map_A013609 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:           5,           2;\n  2:          36,          26,           4;\n  3:         360,         342,          96,          8;\n  4:        4644,        5220,        1980,        304,         16;\n  5:       73548,       92988,       42660,       9000,        880,        32;\n  6:     1382184,     1909656,     1002456,     259560,      35280,      2400,       64;\n  7:    30075624,    44581752,    25923240,    7773192,    1310400,    124992,     6272,     128;\n  8:   743859936,  1168219584,   736849872,  247175712,   48689424,   5806080,   411264,   15872,   256;\n  9: 20613297312, 33988289472, 22920284880, 8409044160, 1867153680, 261981216, 23345280, 1278720, 39168, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(12*n-12)*x+4*x^2)*P(n-2,x) + (6*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*P[n - 2, x] + (6*n - 3 + 4*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*v[i - 2] + (6*n - 3 + 4*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":["basis A013609","D+1"],"edges":["e-0826250557dc6011","e-b0c94102193109a6"],"text":"A225470 \u2192 basis A013609 \u2192 D+1"},"headline":"A225470 \u2192 basis A013609 \u2192 D+1","chain":null}