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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":"e810129e447edea27da99e683dfd44648cf24186e09801d63b9dacb5b668d00a","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-e2ca539e0aa62c2f","role":"sequence","name":"basis_map_A013611 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:    0;\n   2:    4;\n   3:   12,    32;\n   4:   28,   160,    192;\n   5:   60,   512,   1344,    1024;\n   6:  120,  1376,   5568,    9216,    5120;\n   7:  232,  3328,  18240,   47104,   56320,    24576;\n   8:  436,  7520,  52032,  183296,  343040,   319488,   114688;\n   9:  804, 16192, 135360,  605184, 1551360,  2260992,  1720320,  524288;\n  10: 1460, 33632, 329664, 1787904, 5836800, 11673600, 13877248, 8912896, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-16*n+16-(16*n-32)*x)*P(n,x) = -(8*n^2-68*n+120-(80*n-256)*x+128*x^2)*P(n-1,x) + (5*n-15+(24*n-92)*x+(16*n-144)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 68*n + 120 - (80*n - 256)*x + 128*x^2)*P[n - 1, x] + (5*n - 15 + (24*n - 92)*x + (16*n - 144)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(4*n^2 - 16*n + 16 - (16*n - 32)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = (-(8*n^2 - 68*n + 120 - (80*n - 256)*x + 128*x^2)*v[i - 1] + (5*n - 15 + (24*n - 92)*x + (16*n - 144)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(4*n^2 - 16*n + 16 - (16*n - 32)*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-392f49b496f79fbf","steps":["basis A013611","D"],"edges":["e-4d9d32144b57b963","e-10f1169536b0d46e"],"text":"A026729 \u2192 basis A013611 \u2192 D"},"headline":"A026729 \u2192 basis A013611 \u2192 D","chain":null}