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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":"7eb507bf31b67f1cc31a65feeb0986ac537f33543a9a9e176975f42da34296d1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c63944481deaae45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     0;\n   2:     1;\n   3:     5,     2;\n   4:    20,    16,      3;\n   5:    72,    86,     33,     4;\n   6:   244,   388,    225,    56,     5;\n   7:   796,  1580,   1227,   464,    85,     6;\n   8:  2528,  6016,   5856,  2976,   830,   120,    7;\n   9:  7872, 21840,  25560, 16324,  6130,  1350,  161,   8;\n  10: 24144, 76512, 104616, 80488, 38045, 11292, 2051, 208, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^2-5*n+6-(n-2)*x)*P(n,x) = -(2*n^2-22*n+48-(5*n-20)*x+2*x^2)*P(n-1,x) + (12*n-48+(8*n-44)*x+(n-12)*x^2-x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 22*n + 48 - (5*n - 20)*x + 2*x^2)*P[n - 1, x] + (12*n - 48 + (8*n - 44)*x + (n - 12)*x^2 - x^3)*D[P[n - 1, x], x])/(n^2 - 5*n + 6 - (n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 22*n + 48 - (5*n - 20)*x + 2*x^2)*v[i - 1] + (12*n - 48 + (8*n - 44)*x + (n - 12)*x^2 - x^3)*deriv(v[i - 1], x))/(n^2 - 5*n + 6 - (n - 2)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A007318","D"],"edges":["e-3a20d12bade77e7f","e-db405f3564a1ea5d"],"text":"A063967 \u2192 basis A007318 \u2192 D"},"headline":"A063967 \u2192 basis A007318 \u2192 D","chain":null}