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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":"fb2e1a06467e6d35c91d172aa6bd1e50b098a68d0fb85c3b49a78760a4d33f9c","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-387e148a64b95986","role":"sequence","name":"Row reversed version of triangle A201637 (second-order Eulerian triangle).","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,       1;\n  2:      2,       4,        1;\n  3:      6,      22,       11,        1;\n  4:     24,     140,      124,       26,        1;\n  5:    120,    1008,     1428,      536,       57,        1;\n  6:    720,    8136,    16908,    10208,     2022,      120,       1;\n  7:   5040,   73008,   208404,   186620,    60170,     7050,     247,     1;\n  8:  40320,  722592,  2697048,  3361384,  1623020,   315500,   23408,   502,    1;\n  9: 362880, 7830720, 36812808, 60786552, 41267564, 12140500, 1533320, 75292, 1013, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2+2*x)*P(n,x) = -(n^3-4*n^2+5*n-2+(2*n^3-9*n^2+13*n-6)*x+(n^3-14*n+19)*x^2+(3*n^2-8*n+3)*x^3+(2*n-2)*x^4)*P(n-2,x) - ((n^2-3*n+2)*x+(2*n-4)*x^2-(n^2-4*n+3)*x^3-(3*n-7)*x^4-2*x^5)*d/dx P(n-2,x) + (2*n^2-5*n+2+(2*n^2-3*n+2)*x+(5*n-9)*x^2+2*x^3)*P(n-1,x) + ((n-2)*x-(n-4)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 5*n - 2 + (2*n^3 - 9*n^2 + 13*n - 6)*x + (n^3 - 14*n + 19)*x^2 + (3*n^2 - 8*n + 3)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] - ((n^2 - 3*n + 2)*x + (2*n - 4)*x^2 - (n^2 - 4*n + 3)*x^3 - (3*n - 7)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 5*n + 2 + (2*n^2 - 3*n + 2)*x + (5*n - 9)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 2)*x - (n - 4)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 2 + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 5*n - 2 + (2*n^3 - 9*n^2 + 13*n - 6)*x + (n^3 - 14*n + 19)*x^2 + (3*n^2 - 8*n + 3)*x^3 + (2*n - 2)*x^4)*v[i - 2] - ((n^2 - 3*n + 2)*x + (2*n - 4)*x^2 - (n^2 - 4*n + 3)*x^3 - (3*n - 7)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 5*n + 2 + (2*n^2 - 3*n + 2)*x + (5*n - 9)*x^2 + 2*x^3)*v[i - 1] + ((n - 2)*x - (n - 4)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 2 + 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-e2f0f447a502f269","steps":["reverse (shift +1)","D+1"],"edges":["e-5f52c7cc32f7742d","e-91b1367565805f37"],"text":"A201637 \u2192 reverse (shift +1) \u2192 D+1"},"headline":"A201637 \u2192 reverse (shift +1) \u2192 D+1","chain":null}