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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":"015e573a673957b09e57601d5657f9b620fc016add509f73a5722518cbcc47a4","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-7f247edfefc78f8c","role":"sequence","name":"basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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:    4,     3;\n   3:   11,    27,      9;\n   4:   24,   111,    126,     27;\n   5:   50,   333,    711,    513,      81;\n   6:   98,   864,   2736,   3699,    1944,     243;\n   7:  187,  2046,   8631,  17469,   17091,    7047,     729;\n   8:  348,  4557,  24066,  65313,   95904,   73143,   24786,    2187;\n   9:  637,  9711,  61659, 210519,  416016,  476037,  296703,   85293,   6561;\n  10: 1150, 20013, 148536, 612063, 1526526, 2354670, 2198664, 1156923, 288684, 19683;\n","recurrence_sentence":"The row polynomials satisfy (27*n^3-90*n^2+60*n+3+(81*n^2-306*n+225)*x+(81*n^2-216*n+135)*x^2+(81*n-81)*x^3)*P(n,x) = -(54*n^3-423*n^2+822*n-300-(81*n^2+18*n-675)*x+(162*n^2-675*n+783)*x^2-(81*n+243)*x^3-243*x^4)*P(n-1,x) + (45*n^2-105*n+40+(162*n^2-243*n-6)*x+(81*n^2+432*n-423)*x^2+(729*n+27)*x^3+(243*n+567)*x^4+243*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[(-(54*n^3 - 423*n^2 + 822*n - 300 - (81*n^2 + 18*n - 675)*x + (162*n^2 - 675*n + 783)*x^2 - (81*n + 243)*x^3 - 243*x^4)*P[n - 1, x] + (45*n^2 - 105*n + 40 + (162*n^2 - 243*n - 6)*x + (81*n^2 + 432*n - 423)*x^2 + (729*n + 27)*x^3 + (243*n + 567)*x^4 + 243*x^5)*D[P[n - 1, x], x])/(27*n^3 - 90*n^2 + 60*n + 3 + (81*n^2 - 306*n + 225)*x + (81*n^2 - 216*n + 135)*x^2 + (81*n - 81)*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] = (-(54*n^3 - 423*n^2 + 822*n - 300 - (81*n^2 + 18*n - 675)*x + (162*n^2 - 675*n + 783)*x^2 - (81*n + 243)*x^3 - 243*x^4)*v[i - 1] + (45*n^2 - 105*n + 40 + (162*n^2 - 243*n - 6)*x + (81*n^2 + 432*n - 423)*x^2 + (729*n + 27)*x^3 + (243*n + 567)*x^4 + 243*x^5)*deriv(v[i - 1], x))/(27*n^3 - 90*n^2 + 60*n + 3 + (81*n^2 - 306*n + 225)*x + (81*n^2 - 216*n + 135)*x^2 + (81*n - 81)*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-392f49b496f79fbf","steps":["basis A013610","D+1"],"edges":["e-74fcd2d5aef5a8cd","e-fce31363d9bc9ff0"],"text":"A026729 \u2192 basis A013610 \u2192 D+1"},"headline":"A026729 \u2192 basis A013610 \u2192 D+1","chain":null}