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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":"adb0b0b5a1ff2eede51de55e7f1a8aec9122a8e161e2985bbe6a7700c37ec983","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-51fcbaaf07b9691b","role":"sequence","name":"basis_map_A013610 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:     3;\n   3:    15,     18;\n   4:    60,    144,      81;\n   5:   216,    774,     891,     324;\n   6:   732,   3492,    6075,    4536,    1215;\n   7:  2388,  14220,   33129,   37584,   20655,    4374;\n   8:  7584,  54144,  158112,  241056,  201690,   87480,   15309;\n   9: 23616, 196560,  690120, 1322244, 1489590,  984150,  352107,   52488;\n  10: 72432, 688608, 2824632, 6519528, 9244935, 8231868, 4485537, 1364688, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n^2-5*n+6-(3*n-6)*x)*P(n,x) = -(2*n^2-22*n+48-(15*n-60)*x+18*x^2)*P(n-1,x) + (4*n-16+(8*n-44)*x+(3*n-36)*x^2-9*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 22*n + 48 - (15*n - 60)*x + 18*x^2)*P[n - 1, x] + (4*n - 16 + (8*n - 44)*x + (3*n - 36)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(n^2 - 5*n + 6 - (3*n - 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 22*n + 48 - (15*n - 60)*x + 18*x^2)*v[i - 1] + (4*n - 16 + (8*n - 44)*x + (3*n - 36)*x^2 - 9*x^3)*deriv(v[i - 1], x))/(n^2 - 5*n + 6 - (3*n - 6)*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 A013610","D"],"edges":["e-52226b414d15e69f","e-4319e7c4222064e9"],"text":"A063967 \u2192 basis A013610 \u2192 D"},"headline":"A063967 \u2192 basis A013610 \u2192 D","chain":null}