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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":"79c412a9f336fd480da5d5c8d940a19bb4fbe2ab5612c501addcfd1bf333c52c","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:     1;\n   2:     5,      3;\n   3:    21,     33,       9;\n   4:    76,    204,     153,      27;\n   5:   260,    990,    1278,     621,       81;\n   6:   852,   4224,    7821,    6561,     2349,      243;\n   7:  2716,  16608,   40239,   48627,    30051,     8505,     729;\n   8:  8480,  61728,  185184,  293760,   261954,   127818,   29889,    2187;\n   9: 26064, 220176,  788400, 1552284,  1820151,  1282068,  516132,  102789,   6561;\n  10: 79120, 761040, 3168936, 7461072, 10874817, 10080855, 5857515, 2005479, 347733, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-6*n^2-7*n+10+(15*n^2-47*n+32)*x+(9*n^2-15*n+6)*x^2+(9*n-9)*x^3)*P(n,x) = -(6*n^3-48*n^2+82*n+(3*n^2-55*n+120)*x+(18*n^2-84*n+42)*x^2-(9*n+63)*x^3-27*x^4)*P(n-1,x) + (12*n^2-12*n+(24*n^2+36*n-20)*x+(9*n^2+147*n+8)*x^2+(117*n+117)*x^3+(27*n+108)*x^4+27*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^3 - 48*n^2 + 82*n + (3*n^2 - 55*n + 120)*x + (18*n^2 - 84*n + 42)*x^2 - (9*n + 63)*x^3 - 27*x^4)*P[n - 1, x] + (12*n^2 - 12*n + (24*n^2 + 36*n - 20)*x + (9*n^2 + 147*n + 8)*x^2 + (117*n + 117)*x^3 + (27*n + 108)*x^4 + 27*x^5)*D[P[n - 1, x], x])/(3*n^3 - 6*n^2 - 7*n + 10 + (15*n^2 - 47*n + 32)*x + (9*n^2 - 15*n + 6)*x^2 + (9*n - 9)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(6*n^3 - 48*n^2 + 82*n + (3*n^2 - 55*n + 120)*x + (18*n^2 - 84*n + 42)*x^2 - (9*n + 63)*x^3 - 27*x^4)*v[i - 1] + (12*n^2 - 12*n + (24*n^2 + 36*n - 20)*x + (9*n^2 + 147*n + 8)*x^2 + (117*n + 117)*x^3 + (27*n + 108)*x^4 + 27*x^5)*deriv(v[i - 1], x))/(3*n^3 - 6*n^2 - 7*n + 10 + (15*n^2 - 47*n + 32)*x + (9*n^2 - 15*n + 6)*x^2 + (9*n - 9)*x^3)); 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+1"],"edges":["e-52226b414d15e69f","e-3b84a8a6a4072f7c"],"text":"A063967 \u2192 basis A013610 \u2192 D+1"},"headline":"A063967 \u2192 basis A013610 \u2192 D+1","chain":null}