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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":"d11c494ac2a7e01557ade78447378a9e9cfc560ae638ed1fe603da3f5caa7376","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-966c4f25f43e716e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:       7,        3;\n  2:      52,       54,         9;\n  3:     304,      672,       297,        27;\n  4:    1600,     5664,      5400,      1404,        81;\n  5:    7936,    39168,     61488,     34560,      6075,       243;\n  6:   37888,   241152,    539136,    508032,    192780,     24786,       729;\n  7:  176128,  1376256,   4029696,   5495040,   3538080,    979776,     96957,     2187;\n  8:  802816,  7446528,  27076608,  48964608,  46189440,  21881664,   4653936,   367416,    6561;\n  9: 3604480, 38731776, 168652800, 382427136, 483335424, 338722560, 123941664, 20995200, 1358127, 19683;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3-28*n^2+21*n-5+(24*n^2-50*n+19)*x+(12*n-6)*x^2)*P(n,x) = (48*n^3-64*n^2+12*n+4+(108*n^3-120*n^2-107*n+70)*x+(216*n^2-366*n+171)*x^2+(108*n-90)*x^3)*P(n-1,x) - ((24*n^2-8*n-2)*x+(72*n^2+24*n-34)*x^2+(144*n-60)*x^3+72*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((48*n^3 - 64*n^2 + 12*n + 4 + (108*n^3 - 120*n^2 - 107*n + 70)*x + (216*n^2 - 366*n + 171)*x^2 + (108*n - 90)*x^3)*P[n - 1, x] - ((24*n^2 - 8*n - 2)*x + (72*n^2 + 24*n - 34)*x^2 + (144*n - 60)*x^3 + 72*x^4)*D[P[n - 1, x], x])/(12*n^3 - 28*n^2 + 21*n - 5 + (24*n^2 - 50*n + 19)*x + (12*n - 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((48*n^3 - 64*n^2 + 12*n + 4 + (108*n^3 - 120*n^2 - 107*n + 70)*x + (216*n^2 - 366*n + 171)*x^2 + (108*n - 90)*x^3)*v[i - 1] - ((24*n^2 - 8*n - 2)*x + (72*n^2 + 24*n - 34)*x^2 + (144*n - 60)*x^3 + 72*x^4)*deriv(v[i - 1], x))/(12*n^3 - 28*n^2 + 21*n - 5 + (24*n^2 - 50*n + 19)*x + (12*n - 6)*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-c30cb06ad75c5e21","steps":["basis A013610","D+1"],"edges":["e-71859d6a54438fd0","e-4ff4e6a19dfe6c2b"],"text":"A103327 \u2192 basis A013610 \u2192 D+1"},"headline":"A103327 \u2192 basis A013610 \u2192 D+1","chain":null}