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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":"fb31d34038e778cfff405012fcec6b986218294b92bf353200a10d2a8b2b2e65","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-45d3b6c39c71e61d","role":"sequence","name":"basis_map_A008277 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:    1;\n   3:    4,      2;\n   4:   13,     16,      3;\n   5:   39,     88,     39,      4;\n   6:  112,    416,    324,     76,      5;\n   7:  315,   1814,   2208,    884,    130,     6;\n   8:  875,   7544,  13473,   8176,   2010,   204,    7;\n   9: 2414,  30454,  76761,  66208,  24165,  4038,  301,   8;\n  10: 6633, 120572, 417933, 492232, 250030, 61104, 7413, 424, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+2*x+x^2)*P(n-4,x) - (x+x^2)*d/dx P(n-4,x) - (4+4*x+2*x^2)*P(n-3,x) - (2*x+2*x^2)*d/dx P(n-3,x) + (1-x^2)*P(n-2,x) - x^2*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4 + 2*x, P(4,x) = 13 + 16*x + 3*x^2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 4 + 2*x;\nP[4, x_] := 13 + 16*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(2 + 2*x + x^2)*P[n - 4, x] - (x + x^2)*D[P[n - 4, x], x] - (4 + 4*x + 2*x^2)*P[n - 3, x] - (2*x + 2*x^2)*D[P[n - 3, x], x] + (1 - x^2)*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 4 + 2*x; v[4] = 13 + 16*x + 3*x^2; for(n = 5, N, my(i = n); v[i] = -(2 + 2*x + x^2)*v[i - 4] - (x + x^2)*deriv(v[i - 4], x) - (4 + 4*x + 2*x^2)*v[i - 3] - (2*x + 2*x^2)*deriv(v[i - 3], x) + (1 - x^2)*v[i - 2] - x^2*deriv(v[i - 2], x) + (3 + 2*x)*v[i - 1] + x*deriv(v[i - 1], 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 A008277","D"],"edges":["e-08527b1121a7be71","e-ede66256a5fd5e8b"],"text":"A026729 \u2192 basis A008277 \u2192 D"},"headline":"A026729 \u2192 basis A008277 \u2192 D","chain":null}