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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":"9dec6e98f20c138c95f794c480869cdf4f32437f80fabcf05ee01a3752f1583b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d6be23d68d7706ed","role":"sequence","name":"basis_map_A013609 of A Riordan array of coordination sequences.","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:    19,     20,      4;\n  3:    63,    118,     60,       8;\n  4:   193,    536,    480,     160,      16;\n  5:   565,   2098,   2784,    1616,     400,     32;\n  6:  1603,   7484,  13332,   11424,    4880,    960,     64;\n  7:  4447,  25062,  56364,   64856,   40720,  13728,   2240,   128;\n  8: 12129,  80176, 218464,  317888,  267680, 132096,  36736,  5120,   256;\n  9: 32645, 247778, 794400, 1403296, 1494688, 987456, 400512, 94720, 11520, 512;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-6*n-2+(2*n^3-17)*x+(4*n^2-4*n-11)*x^2+(2*n^2+3*n+3)*x^3+(2*n+2)*x^4)*P(n,x) = -(4*n^3-24*n^2+18*n+22+(6*n^3-42*n^2+8*n+111)*x-(50*n-135)*x^2+(6*n^2-23*n-1)*x^3-(2*n+34)*x^4-8*x^5)*P(n-1,x) + (4*n^2-4*n-4+(16*n^2-8*n-26)*x+(18*n^2+18*n-48)*x^2+(4*n^2+48*n-5)*x^3+(26*n+51)*x^4+(4*n+28)*x^5+4*x^6)*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[(-(4*n^3 - 24*n^2 + 18*n + 22 + (6*n^3 - 42*n^2 + 8*n + 111)*x - (50*n - 135)*x^2 + (6*n^2 - 23*n - 1)*x^3 - (2*n + 34)*x^4 - 8*x^5)*P[n - 1, x] + (4*n^2 - 4*n - 4 + (16*n^2 - 8*n - 26)*x + (18*n^2 + 18*n - 48)*x^2 + (4*n^2 + 48*n - 5)*x^3 + (26*n + 51)*x^4 + (4*n + 28)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(4*n^2 - 6*n - 2 + (2*n^3 - 17)*x + (4*n^2 - 4*n - 11)*x^2 + (2*n^2 + 3*n + 3)*x^3 + (2*n + 2)*x^4)]];\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] = (-(4*n^3 - 24*n^2 + 18*n + 22 + (6*n^3 - 42*n^2 + 8*n + 111)*x - (50*n - 135)*x^2 + (6*n^2 - 23*n - 1)*x^3 - (2*n + 34)*x^4 - 8*x^5)*v[i - 1] + (4*n^2 - 4*n - 4 + (16*n^2 - 8*n - 26)*x + (18*n^2 + 18*n - 48)*x^2 + (4*n^2 + 48*n - 5)*x^3 + (26*n + 51)*x^4 + (4*n + 28)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(4*n^2 - 6*n - 2 + (2*n^3 - 17)*x + (4*n^2 - 4*n - 11)*x^2 + (2*n^2 + 3*n + 3)*x^3 + (2*n + 2)*x^4)); 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-ec35d1145e9a1211","steps":["basis A013609","D+1"],"edges":["e-7383986e8ecff3a4","e-dd6b3c573b522abc"],"text":"A113413 \u2192 basis A013609 \u2192 D+1"},"headline":"A113413 \u2192 basis A013609 \u2192 D+1","chain":null}