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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":"95aebb85ce90bae0c4bb1de00286459ce5572c0ee4c1bc21240ba6ddc419784c","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-a9307f854ca152e2","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 3+x).","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:     19,      18;\n   4:     93,     174,      81;\n   5:    407,    1136,    1053,      324;\n   6:   1674,    6210,    8613,     5292,     1215;\n   7:   6618,   30624,   56541,    52056,    23895,     4374;\n   8:  25455,  141138,  325359,   399108,   274725,   100602,   15309;\n   9:  95953,  620004, 1713675,  2626204,  2409795,  1323864,  403137,   52488;\n  10: 356151, 2627778, 8468109, 15566232, 17851260, 13077126, 5975613, 1557144, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2-36*n+60-(27*n-54)*x)*P(n,x) = -(2*n^2-57*n+286-(45*n-432)*x+162*x^2)*P(n-1,x) + (13*n-143+(22*n-359)*x+(9*n-297)*x^2-81*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 - 57*n + 286 - (45*n - 432)*x + 162*x^2)*P[n - 1, x] + (13*n - 143 + (22*n - 359)*x + (9*n - 297)*x^2 - 81*x^3)*D[P[n - 1, x], x])/(3*n^2 - 36*n + 60 - (27*n - 54)*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 - 57*n + 286 - (45*n - 432)*x + 162*x^2)*v[i - 1] + (13*n - 143 + (22*n - 359)*x + (9*n - 297)*x^2 - 81*x^3)*deriv(v[i - 1], x))/(3*n^2 - 36*n + 60 - (27*n - 54)*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-5d39164c4108e043","steps":["basis A007318","D"],"edges":["e-0e2d8787d6ee9a58","e-cd311ed7b1755df7"],"text":"A099097 \u2192 basis A007318 \u2192 D"},"headline":"A099097 \u2192 basis A007318 \u2192 D","chain":null}