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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":"67fd804c5afdd1b9767468745b221866541a027a91181a90ffa2c5a1d6445677","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-44370b381098ad54","role":"sequence","name":"basis_map_A013610 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:     0;\n  1:     3;\n  2:    18,     18;\n  3:    69,    162,      81;\n  4:   228,    864,     972,     324;\n  5:   699,   3672,    6642,    4860,    1215;\n  6:  2046,  13770,   34668,   40500,   21870,    4374;\n  7:  5805,  47682,  154467,  251100,  215055,   91854,   15309;\n  8: 16104, 156240,  620136, 1299240, 1545480, 1041012,  367416,   52488;\n  9: 43923, 491616, 2312388, 5947992, 9119790, 8511804, 4715172, 1417176, 177147;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2-36*n+24+(9*n^3-18*n^2-36*n+45)*x-(27*n^2-27)*x^2)*P(n,x) = -(12*n^3-96*n^2+168*n-48+(27*n^3-315*n^2+675*n-144)*x-(189*n^2-729*n+54)*x^2+243*n*x^3)*P(n-1,x) + (8*n^2-24*n+8+(48*n^2-168*n+48)*x+(81*n^2-387*n+81)*x^2+(27*n^2-324*n+27)*x^3-81*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n^3 - 96*n^2 + 168*n - 48 + (27*n^3 - 315*n^2 + 675*n - 144)*x - (189*n^2 - 729*n + 54)*x^2 + 243*n*x^3)*P[n - 1, x] + (8*n^2 - 24*n + 8 + (48*n^2 - 168*n + 48)*x + (81*n^2 - 387*n + 81)*x^2 + (27*n^2 - 324*n + 27)*x^3 - 81*n*x^4)*D[P[n - 1, x], x])/(12*n^2 - 36*n + 24 + (9*n^3 - 18*n^2 - 36*n + 45)*x - (27*n^2 - 27)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^3 - 96*n^2 + 168*n - 48 + (27*n^3 - 315*n^2 + 675*n - 144)*x - (189*n^2 - 729*n + 54)*x^2 + 243*n*x^3)*v[i - 1] + (8*n^2 - 24*n + 8 + (48*n^2 - 168*n + 48)*x + (81*n^2 - 387*n + 81)*x^2 + (27*n^2 - 324*n + 27)*x^3 - 81*n*x^4)*deriv(v[i - 1], x))/(12*n^2 - 36*n + 24 + (9*n^3 - 18*n^2 - 36*n + 45)*x - (27*n^2 - 27)*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-ec35d1145e9a1211","steps":["basis A013610","D"],"edges":["e-3d8a64e4b99d997d","e-4b14084d626165cd"],"text":"A113413 \u2192 basis A013610 \u2192 D"},"headline":"A113413 \u2192 basis A013610 \u2192 D","chain":null}