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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":"95243b567ca06f6023711f1b5b97874a207fe71afe0a052d3afe9d58584a7e62","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-3a1bf08f9f4a4163","role":"sequence","name":"basis_map_A013610 of Pascal-(1,2,1) array read by antidiagonals.","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:    72,     180,      81;\n  4:   264,    1044,    1134,      324;\n  5:   900,    4896,    8748,     5832,     1215;\n  6:  2952,   20448,   51192,    56376,    26730,     4374;\n  7:  9408,   79200,  255960,   399168,   311040,   113724,   15309;\n  8: 29376,  291168, 1154736,  2344464,  2595240,  1548396,  459270,   52488;\n  9: 90288, 1029888, 4844448, 12153888, 17598060, 14906592, 7164612, 1784592, 177147;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(3*n^2-3*n)*x-(3*n-3)*x^2)*P(n,x) = -(6*n^2-26*n+16+(15*n^2-87*n+66)*x-(42*n-66)*x^2+18*x^3)*P(n-1,x) + (4*n-4+(24*n-28)*x+(39*n-63)*x^2+(9*n-48)*x^3-9*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[(-(6*n^2 - 26*n + 16 + (15*n^2 - 87*n + 66)*x - (42*n - 66)*x^2 + 18*x^3)*P[n - 1, x] + (4*n - 4 + (24*n - 28)*x + (39*n - 63)*x^2 + (9*n - 48)*x^3 - 9*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (3*n^2 - 3*n)*x - (3*n - 3)*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] = (-(6*n^2 - 26*n + 16 + (15*n^2 - 87*n + 66)*x - (42*n - 66)*x^2 + 18*x^3)*v[i - 1] + (4*n - 4 + (24*n - 28)*x + (39*n - 63)*x^2 + (9*n - 48)*x^3 - 9*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (3*n^2 - 3*n)*x - (3*n - 3)*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-f25cd955a876cde3","steps":["basis A013610","D"],"edges":["e-ce7b55dc0173f12e","e-5375dcb77da2f74b"],"text":"A081577 \u2192 basis A013610 \u2192 D"},"headline":"A081577 \u2192 basis A013610 \u2192 D","chain":null}