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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":"fe00e41aff4f5e17d8865483fa684633a55a1ae2e35436c100dfdd984f616228","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-0515531ee2138839","role":"sequence","name":"reversal of Riordan array (1, 3+3x).","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:     3;\n  2:    12,      3;\n  3:    45,     18;\n  4:   162,     99,     9;\n  5:   567,    486,    81;\n  6:  1944,   2187,   567,    27;\n  7:  6561,   9234,  3402,   324;\n  8: 21870,  37179, 18225,  2754,   81;\n  9: 72171, 144342, 89667, 19440, 1215;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n+(4*n^2-7*n)*x+4*n*x^2)*P(n,x) = (3*n^3+3*n^2-6*n+(2*n^3+10*n^2-21*n)*x+(8*n^2-2*n+6)*x^2+(8*n-8)*x^3)*P(n-1,x) - ((3*n^2+6*n)*x+(4*n^2+20*n+3)*x^2+(16*n+16)*x^3+16*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[((3*n^3 + 3*n^2 - 6*n + (2*n^3 + 10*n^2 - 21*n)*x + (8*n^2 - 2*n + 6)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - ((3*n^2 + 6*n)*x + (4*n^2 + 20*n + 3)*x^2 + (16*n + 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - n + (4*n^2 - 7*n)*x + 4*n*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] = ((3*n^3 + 3*n^2 - 6*n + (2*n^3 + 10*n^2 - 21*n)*x + (8*n^2 - 2*n + 6)*x^2 + (8*n - 8)*x^3)*v[i - 1] - ((3*n^2 + 6*n)*x + (4*n^2 + 20*n + 3)*x^2 + (16*n + 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - n + (4*n^2 - 7*n)*x + 4*n*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-016d7724d51f6416","steps":["reverse","D+1"],"edges":["e-450f652a77090025","e-b79bbe32143ed79b"],"text":"A099093 \u2192 reverse \u2192 D+1"},"headline":"A099093 \u2192 reverse \u2192 D+1","chain":null}