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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":"37315f52237649fbc36cbb29f361314dda96a2877a190867e6392590a71c10bd","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-45ec3349543b0d37","role":"sequence","name":"Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.","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:       2,       1;\n  2:       5,       5,        1;\n  3:      18,      26,       10,        1;\n  4:      84,     160,       95,       19,       1;\n  5:     480,    1140,      930,      304,      36,       1;\n  6:    3240,    9240,     9660,     4508,     917,      69,      1;\n  7:   25200,   84000,   107520,    66024,   19866,    2694,    134,     1;\n  8:  221760,  846720,  1285200,   984816,  399924,   82920,   7835,   263,   1;\n  9: 2177280, 9374400, 16480800, 15190560, 7842240, 2248380, 335250, 22748, 520, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2*x)*P(n,x) = -(n^3-2*n^2-n+2+(2*n^3-n^2-8*n+7)*x+(n^3+4*n^2-11*n+6)*x^2+(3*n^2-2*n-1)*x^3+(2*n-2)*x^4)*P(n-2,x) + ((n^2-1)*x+(2*n^2+3*n-3)*x^2+(n^2+6*n-1)*x^3+(3*n+3)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2-n-1+(2*n^2+3*n-4)*x+(5*n-3)*x^2+2*x^3)*P(n-1,x) - (n*x+(n+2)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 2*n^2 - n + 2 + (2*n^3 - n^2 - 8*n + 7)*x + (n^3 + 4*n^2 - 11*n + 6)*x^2 + (3*n^2 - 2*n - 1)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + ((n^2 - 1)*x + (2*n^2 + 3*n - 3)*x^2 + (n^2 + 6*n - 1)*x^3 + (3*n + 3)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - n - 1 + (2*n^2 + 3*n - 4)*x + (5*n - 3)*x^2 + 2*x^3)*P[n - 1, x] - (n*x + (n + 2)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 2*n^2 - n + 2 + (2*n^3 - n^2 - 8*n + 7)*x + (n^3 + 4*n^2 - 11*n + 6)*x^2 + (3*n^2 - 2*n - 1)*x^3 + (2*n - 2)*x^4)*v[i - 2] + ((n^2 - 1)*x + (2*n^2 + 3*n - 3)*x^2 + (n^2 + 6*n - 1)*x^3 + (3*n + 3)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - n - 1 + (2*n^2 + 3*n - 4)*x + (5*n - 3)*x^2 + 2*x^3)*v[i - 1] - (n*x + (n + 2)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2*x)); 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-45ec3349543b0d37","steps":["D+1"],"edges":["e-81cc9d1fcd90d4f7"],"text":"A130850 \u2192 D+1"},"headline":"A130850 \u2192 D+1","chain":null}