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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":"e8d67d27eaf4012512496f9deb5928acb4ccf45561e4aea96a13f40ad98c86c4","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-3e465972400fd143","role":"sequence","name":"shift_x_plus_one of Triangular array of Motzkin polynomial coefficients.","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:    1;\n  2:    3,    1;\n  3:    7,    3;\n  4:   19,   14,    2;\n  5:   51,   50,   10;\n  6:  141,  180,   60,    5;\n  7:  393,  616,  280,   35;\n  8: 1107, 2072, 1232,  252,  14;\n  9: 3139, 6840, 5040, 1428, 126;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2-n-6+(4*n^2+5*n-6)*x+(4*n+8)*x^2)*P(n,x) = (5*n^3+5*n^2-11*n-2+(4*n^3+20*n^2-19*n-2)*x+(16*n^2+8*n-12)*x^2+(16*n-16)*x^3)*P(n-1,x) - (6*n^2+18*n+12+(14*n^2+66*n+58)*x+(8*n^2+80*n+110)*x^2+(32*n+96)*x^3+32*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[((5*n^3 + 5*n^2 - 11*n - 2 + (4*n^3 + 20*n^2 - 19*n - 2)*x + (16*n^2 + 8*n - 12)*x^2 + (16*n - 16)*x^3)*P[n - 1, x] - (6*n^2 + 18*n + 12 + (14*n^2 + 66*n + 58)*x + (8*n^2 + 80*n + 110)*x^2 + (32*n + 96)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - n - 6 + (4*n^2 + 5*n - 6)*x + (4*n + 8)*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] = ((5*n^3 + 5*n^2 - 11*n - 2 + (4*n^3 + 20*n^2 - 19*n - 2)*x + (16*n^2 + 8*n - 12)*x^2 + (16*n - 16)*x^3)*v[i - 1] - (6*n^2 + 18*n + 12 + (14*n^2 + 66*n + 58)*x + (8*n^2 + 80*n + 110)*x^2 + (32*n + 96)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 - n - 6 + (4*n^2 + 5*n - 6)*x + (4*n + 8)*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-169bd483754ab390","steps":["basis A007318","D+1"],"edges":["e-5fd319f73e442e4d","e-063135293394c29d"],"text":"A055151 \u2192 basis A007318 \u2192 D+1"},"headline":"A055151 \u2192 basis A007318 \u2192 D+1","chain":null}