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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":"390e295faf65267315ae411023d12d847201559a1aacba86098e52e01df15f5c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b60ac0f95ba69845","role":"sequence","name":"derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n 0: 0;\n 1: 2;\n 2: 22, 36, 12;\n 3: 60, 270, 360, 180, 30;\n 4: 116, 864, 2340, 2880, 1740, 504, 56;\n 5: 190, 1980, 8220, 17460, 20550, 13860, 5320, 1080, 90;\n 6: 282, 3780, 21240, 65160, 119670, 136836, 99120, 45360, 12690, 1980, 132;\n 7: 392, 6426, 45612, 183960, 465990, 776286, 869848, 662256, 342090, 117810, 25872, 3276, 182;\n 8: 520, 10080, 86520, 433440, 1407840, 3119760, 4844840, 5348160, 4223520, 2383920, 951720, 262080, 47320, 5040, 240;\n 9: 666, 14904, 150120, 899640, 3574620, 9942912, 19951848, 29383560, 32065470, 26040960, 15728328, 7017192, 2276820, 521640, 79920, 7344, 306;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2-25*n+16+(12*n^2-30*n+18)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (9*n^2-7*n+(39*n^2-27*n)*x+(49*n^2-27*n)*x^2+(24*n^2-12*n)*x^3+(4*n^2-2*n)*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 7*n + (39*n^2 - 27*n)*x + (49*n^2 - 27*n)*x^2 + (24*n^2 - 12*n)*x^3 + (4*n^2 - 2*n)*x^4)*P[n - 1, x])/(9*n^2 - 25*n + 16 + (12*n^2 - 30*n + 18)*x + (4*n^2 - 10*n + 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 7*n + (39*n^2 - 27*n)*x + (49*n^2 - 27*n)*x^2 + (24*n^2 - 12*n)*x^3 + (4*n^2 - 2*n)*x^4)*v[i - 1])/(9*n^2 - 25*n + 16 + (12*n^2 - 30*n + 18)*x + (4*n^2 - 10*n + 6)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b30e7a5b4198051a","steps":["D","D"],"edges":["e-5fb206b09638d5b7","e-3cf3bbceb1a10b03"],"text":"A272866 \u2192 D \u2192 D"},"headline":"A272866 \u2192 D \u2192 D","chain":null}