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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":"a42d28444b06f95d92b56f96aa00f6c6078f040641ab4ac2b36e8d67dad5ad3a","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-37ca9f2607503703","role":"sequence","name":"basis_map_A013609 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:     2;\n  2:    12,      8;\n  3:    48,     80,      24;\n  4:   176,    464,     336,      64;\n  5:   600,   2176,    2592,    1152,     160;\n  6:  1968,   9088,   15168,   11136,    3520,     384;\n  7:  6272,  35200,   75840,   78848,   40960,    9984,    896;\n  8: 19584, 129408,  342144,  463104,  341760,  135936,  26880,  2048;\n  9: 60192, 457728, 1435392, 2400768, 2317440, 1308672, 419328, 69632, 4608;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3+(3*n^2-3*n)*x-(2*n-2)*x^2)*P(n,x) = -(9*n^2-39*n+24+(15*n^2-87*n+66)*x-(28*n-44)*x^2+8*x^3)*P(n-1,x) + (9*n-9+(36*n-42)*x+(39*n-63)*x^2+(6*n-32)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 39*n + 24 + (15*n^2 - 87*n + 66)*x - (28*n - 44)*x^2 + 8*x^3)*P[n - 1, x] + (9*n - 9 + (36*n - 42)*x + (39*n - 63)*x^2 + (6*n - 32)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(3*n - 3 + (3*n^2 - 3*n)*x - (2*n - 2)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 39*n + 24 + (15*n^2 - 87*n + 66)*x - (28*n - 44)*x^2 + 8*x^3)*v[i - 1] + (9*n - 9 + (36*n - 42)*x + (39*n - 63)*x^2 + (6*n - 32)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(3*n - 3 + (3*n^2 - 3*n)*x - (2*n - 2)*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 A013609","D"],"edges":["e-144ed6eafdc58374","e-185268329591ff74"],"text":"A081577 \u2192 basis A013609 \u2192 D"},"headline":"A081577 \u2192 basis A013609 \u2192 D","chain":null}