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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":"bc24e9633c9b31e17126e79c216c18c4fdca152bbf0188a57c8b03965239cc78","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-78b19c28b9f05152","role":"sequence","name":"veronese_section_2_0 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: 1;\n 2: 11, 2;\n 3: 30, 60, 3;\n 4: 58, 390, 174, 4;\n 5: 95, 1370, 2055, 380, 5;\n 6: 141, 3540, 11967, 7080, 705, 6;\n 7: 196, 7602, 46599, 62132, 19005, 1176, 7;\n 8: 260, 14420, 140784, 346060, 234640, 43260, 1820, 8;\n 9: 333, 25020, 357462, 1425132, 1781415, 714924, 87570, 2664, 9;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2-25*n+16-(13*n^2-25*n+12)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (9*n^2-7*n+(59*n^2-193*n)*x-(55*n^2-107*n)*x^2+(12*n^2-22*n)*x^3)*P(n-1,x) + (18*n*x-134*n*x^2+74*n*x^3-8*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 7*n + (59*n^2 - 193*n)*x - (55*n^2 - 107*n)*x^2 + (12*n^2 - 22*n)*x^3)*P[n - 1, x] + (18*n*x - 134*n*x^2 + 74*n*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(9*n^2 - 25*n + 16 - (13*n^2 - 25*n + 12)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 7*n + (59*n^2 - 193*n)*x - (55*n^2 - 107*n)*x^2 + (12*n^2 - 22*n)*x^3)*v[i - 1] + (18*n*x - 134*n*x^2 + 74*n*x^3 - 8*n*x^4)*deriv(v[i - 1], x))/(9*n^2 - 25*n + 16 - (13*n^2 - 25*n + 12)*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":["Veronese 2","D"],"edges":["e-864f7f13d69989fc","e-4afe1be6d3951749"],"text":"A272866 \u2192 Veronese 2 \u2192 D"},"headline":"A272866 \u2192 Veronese 2 \u2192 D","chain":null}