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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":"19975f4b29ac0eaf69d1668bbea52cb3d9148c59a38a6168b982f4c9dfdf4a1e","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-5d7c3f4772a9d8bb","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.","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:      11,       5;\n  4:      52,      38,       5;\n  5:     300,     302,      61;\n  6:    2040,    2644,     845,     61;\n  7:   15960,   25452,   11421,   1385;\n  8:  141120,  268128,  157368,  30108,  1385;\n  9: 1391040, 3074400, 2248200, 610444, 50521;\n","recurrence_sentence":"The row polynomials satisfy (n+4*x)*P(n,x) = -(n^3-3*n^2-n+3+(2*n^3-2*n^2-18*n+18)*x+(n^3+7*n^2-32*n+26)*x^2+(6*n^2-10*n+4)*x^3+(8*n-16)*x^4)*P(n-2,x) + ((2*n^2-2)*x+(4*n^2+10*n-10)*x^2+(2*n^2+22*n-4)*x^3+(12*n+20)*x^4+16*x^5)*d/dx P(n-2,x) + (2*n^2-2*n-1+(2*n^2+4*n-12)*x+(10*n-18)*x^2+8*x^3)*P(n-1,x) - (2*n*x+(2*n+8)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 - n + 3 + (2*n^3 - 2*n^2 - 18*n + 18)*x + (n^3 + 7*n^2 - 32*n + 26)*x^2 + (6*n^2 - 10*n + 4)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] + ((2*n^2 - 2)*x + (4*n^2 + 10*n - 10)*x^2 + (2*n^2 + 22*n - 4)*x^3 + (12*n + 20)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (2*n^2 - 2*n - 1 + (2*n^2 + 4*n - 12)*x + (10*n - 18)*x^2 + 8*x^3)*P[n - 1, x] - (2*n*x + (2*n + 8)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 - n + 3 + (2*n^3 - 2*n^2 - 18*n + 18)*x + (n^3 + 7*n^2 - 32*n + 26)*x^2 + (6*n^2 - 10*n + 4)*x^3 + (8*n - 16)*x^4)*v[i - 2] + ((2*n^2 - 2)*x + (4*n^2 + 10*n - 10)*x^2 + (2*n^2 + 22*n - 4)*x^3 + (12*n + 20)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (2*n^2 - 2*n - 1 + (2*n^2 + 4*n - 12)*x + (10*n - 18)*x^2 + 8*x^3)*v[i - 1] - (2*n*x + (2*n + 8)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n + 4*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-d8312bb744732845","steps":["\u03b3 (scale 1/4)","x \u21a6 x+1","D+1"],"edges":["e-16125ab4d60fcfd7","e-42ac1b21fb8d0456","e-19770811effd7f7c"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}